A comprehensive statistical study on gamma-ray bursts

In order to obtain an overview of the gamma-ray bursts (GRBs), we need a full sample. In this paper, we collected 6289 GRBs (from GRB 910421 to GRB 160509A) from the literature, including prompt emission, afterglow and host galaxy properties. We hope to use this large sample to reveal the intrinsic properties of GRB. We have listed all the data in machine readable tables, including the properties of the GRBs, correlation coefficients and linear regression results of two arbitrary parameters, and linear regression results of any three parameters. These machine readable tables could be used as a data reservoir for further studies on the classifications or correlations. One may find some intrinsic properties from these statistical results. With this comprehensive table, it is possible to find relations between different parameters, and to classify the GRBs into different kinds of sub-groups. With the completion, it may reveal the nature of GRBs and may be used as tools like pseudo-redshift indicators, standard candles, etc. All the machine readable data and statistical results are available on the website of the journal.


INTRODUCTION
Gamma-ray bursts (GRBs) were firstly detected in 1967 (Klebesadel et al. 1973) by the Vela Satellite Network. They have been intensely studied since 1990s, especially after the operation of Burst and Transient Source Experiment (BATSE) onboard Compton Gamma Ray Observatory (CGRO) (Meegan et al. 1992). For a single event, the fluence is between 10 −7 and 10 −5 ergs cm −2 , and the isotropic energy is from about 10 48 to 10 55 ergs (Nakar 2007;Gehrels & Razzaque 2013;Berger 2014;Kumar & Zhang 2015). So GRBs are the most violent and energetic stellar explosions known to humankind in distant galaxies (Kumar & Zhang 2015). The nature of GRBs is still under discovered even though there are many researches have been going on for many years. Fireball model is one of historical models to explain the mechanism of GRBs (Rees & Meszaros 1992;Piran et al. 1993;Wijers et al. 1997;Mészáros 1998Mészáros , 2006, which implies GRBs are produced by highly relativistic and collimated jets. Interaction of blobs in the jet is believed to produce the prompt emission, and the interaction of the jet with the ambient material produces the multi-wavelength afterglow (X-ray, optical and sometimes also radio). For a clear understanding of the current situation for GRBs, some recent reviews could be read, such as Piran (1999Piran ( , 2004; Zhang & Mészáros (2004); Mészáros (2006); Zhang (2007); Kumar & Zhang (2015) for the general physics, Nakar (2007); Berger (2014) for short GRBs, Gehrels et al. (2009)  In order to explain the physics of GRBs, many attempts have been made to describe the spectra in different physical frameworks, and significant progress is achieved. There are two main frameworks. One is internal or external shocks, the emissions are assumed to be non-thermal intrinsically (Katz 1994;Rees & Meszaros 1994;Tavani 1996;Sari et al. 1998;Gao et al. 2015b). Another is photospheric emission, which is predicted to occur in the "fireball" model (Mészáros & Rees 2000;Rees & Mészáros 2005;Pe'er et al. 2007;Thompson et al. 2007). Racz et al. (2017) found the synchrotron radiation is significant in Fermi /Gamma-ray Burst Monitor (GBM) spectra. Sub-photospheric dissipation (Rees & Mészáros 2005;Pe'er et al. 2006a;Giannios 2006;Chhotray & Lazzati 2015) was suggested as the emission mechanism in some bursts which can broaden the spectrum. Ahlgren et al. (2015) firstly provided a full physical model, DREAM (Dissipation with Radiative Emission as A table Model), which is based on sub-photospheric dissipation, and gave acceptable fits to GRB 090618 and GRB 100724B for more details. If we take sub-photospheric dissipation and/or high latitude effects (Lundman et al. 2013), the photospheric model can account for a large diversity of spectra. Recently, the spectra observations of GRBs show a mixture of thermal and non-thermal spectra (Ryde 2005;Ryde & Pe'er 2009;Guiriec et al. 2010;Nappo et al. 2017), which implies that there is an interplay between different emission mechanisms. The exist of GRBs subgroups might be the consequence of different emission mechanisms (Bégué & Burgess 2016). Acuner & Ryde (2017) searched the full Fermi /GBM catalogue using Gaussian Mixture Models to cluster bursts according to their low energy photon index of Band model (Band et al. 1993) (α band ), high energy photon index of Band model (β band ), spectral peak energy of Band model (E p,band ), fluence (F g ) and duration of 5% to 95% γ-ray fluence (T 90 ), in order to divide bursts into photospheric origin and synchrotron origin. They thought both emission from the photosphere and optically-thin synchrotron radiation are operating but different emissions dominate differently for individual GRBs. They found 1/3 of the bursts are consistent with synchrotron radiation, and 2/3 of all bursts are consistent with photospheric emission. Beside the two main groups, they also found subgroups. It is maybe due to the dissipation pattern in the jet, alternatively due to whether the jet is dominated by thermal or magnetic energy, or due to the viewing angle (Acuner & Ryde 2017). For the afterglows of GRBs, we can also find thermal and non-thermal components. Up to now, there are totally 16 GRBs reported to have thermal components in the X-ray light curves (LCs) (Campana et al. 2006;Page et al. 2011;Thöne et al. 2011;Starling et al. 2011;Sparre & Starling 2012;Starling et al. 2012;Friis & Watson 2013;Nappo et al. 2017). Valan et al. (2017) reported six detections of additional thermal components in the GRBs' early X-ray afterglows. A cocoon breaking out from a thick wind or late prompt emission maybe the explanation for the additional thermal components, which may be hidden by bright afterglows in the majority of GRBs. That is why small part of the afterglows detected with additional thermal components (Valan et al. 2017). Some other possible explanations are that shock break-out is appealing for GRBs with associated supernova (Campana et al. 2006), or the cocoon that surrounds the jet (Mészáros & Rees 2001;Pe'er et al. 2006b;Ghisellini et al. 2007;Starling et al. 2012), or the jet itself (Friis & Watson 2013;Irwin & Chevalier 2016;Nappo et al. 2017).
With accumulated data, statistical study may reveal the underlying physics. It can be roughly split into classification and correlation seeking. Nowadays, the classification of GRBs is still in suspense. Traditionally, they are Statistics on GRBs 3 grouped into short GRBs (SGRBs) and long GRBs (LGRBs) depending on T 90 smaller or greater than 2 s, where T 90 is the time difference between the 95th and 5th percentile of the total counts, which is often taken as the typical duration of a GRB. Kouveliotou et al. (1993) analysed the log T 90 distribution of 222 BATSE GRBs. They found that the distribution is a bimodal form, and the dividing line is about 2 s. However, T 90 is a function of energy band. Consequently, this value is detector dependent (Bromberg et al. 2013;Resmi 2017). Bromberg et al. (2013) calculated a useful threshold duration that separates collapsars (long) from non-collapsars (short), T 90 = 3.1 ± 0.5 s in BATSE, T 90 = 1.7 +0. 4 −0.6 s in Fermi GBM, T 90 = 0.8 ± 0.3 s in Swift.
With an increasing number of instruments applied for detecting GRBs, such as BeppoSAX/GRBM (40 -700 keV) (Boella et al. 1997), Konus-Wind (10 keV -10 MeV) (Aptekar et al. 1995), HETE-2 (6 -400 keV) (Ricker et al. 2003), INTEGRAL (15 -200 keV) (Mereghetti et al. 2003), Swift/XRT (0.2 -10 keV) (Gehrels et al. 2004;Burrows et al. 2005), Swift/BAT (15 -150 keV) (Gehrels et al. 2004; Barthelmy et al. 2005), Polar (10 -500 keV) (Produit et al. 2018), HXMT (10 -500 keV) (Zhang et al. 2014b), Fermi /LAT (20 MeV -300 GeV) (GLAST Facility Science Team et al. 1999;Ackermann et al. 2012), Fermi /GBM (8 keV -40 MeV) (GLAST Facility Science Team et al. 1999;Meegan et al. 2009), the Alpha Magnetic Spectrometer (AMS-02) (0.5 GeV -2 TeV) (Kounine 2012), the DArk Matter Particle Explorer (DAMPE, or called Wukong) (5 GeV -10 TeV) (Chang et al. 2017), we can have much better quality data for analyzing. Some researches argued that there maybe exits a third class of GRBs (e.g. de Ugarte Postigo et al. 2011). Tsutsui et al. (2013a identified subclasses of LGRBs with cumulative light curve of the prompt emission. Different subclasses have different Fundamental Planes, which is a correlation between L pk , E pk and T L (T L = E iso /L pk ) (Tsutsui et al. 2011). Forthermore, Tsutsui & Shigeyama (2014) confirmed that there do exist the third class in GRBs in addition to SGRBs and LGRBs. The classification method is based on two properties both quantified with light curve shapes of the prompt emission: the absolute deviation from the constant luminosity of their cumulative light curve, and the ratio of the mean counts to the maximum counts (Tsutsui et al. 2013a;Tsutsui & Shigeyama 2014). A new class of GRBs with thousands of seconds of duration known as ultralong bursts have been discovered (Stratta et al. 2013;Levan et al. 2014;Horesh et al. 2015;Gompertz & Fruchter 2017). The possible candidates for ultra-long bursts origin are blue supergiant collapsars, magnetars, and white dwarf tidal disruption events caused by massive black holes (Stratta et al. 2013;Ioka et al. 2016;Perets et al. 2016;Gompertz & Fruchter 2017). However, Evans et al. (2014) considered and rejected the possibility that ultra-long GRB 130925A was form of tidal disruption event, and instead showed that if the circumburst density around ultra-long GRB 130925A is low, the long duration of the burst and faint external shock emission are naturally explained. On the contrary, Gompertz & Fruchter (2017) suggested a high density circumburst environment for ultra-long GRB 111209A, which was powered by the spin-down of a highly magnetized millisecond pulsar. Norris & Bonnell (2006) found the existence of an intermediate class (IC) or SGRBs with Extended Emission (SGRBsEE), which shows mixed properties between SGRBs and LGRBs. Some models have been proposed to explain SGRBsEE, like fallback accretion onto a newborn magnetar citepRowlinson2012, Rowlinson2014,Rea2015,Gibson2017,Stratta2018. Horváth (1998 and Horváth (2002) investigated 797 and 1929 BASTE GRBs respectively, and found three-Gaussian (3-G) distribution is better than two-Gaussian (2-G) statistically. Similar results can be found in the analysis of BeppoSAX (Horváth 2009), Swift /BAT (Horváth et al. 2008;Horváth & Tóth 2016), Fermi /GBM (Tarnopolski 2015) etc. The analyses method is χ 2 fitting (Horváth 1998;Tarnopolski 2015) or Maximum Likelihood method (Horváth 2002;Horváth et al. 2008;Horváth 2009), and the rejection probability is less than 0.5%. Zitouni et al. (2015) investigated 248 Swift/BAT GRBs and also prefers to 3-G distribution, but prefers to 2-G distribution for BATSE bursts. It was suggested by Zitouni et al. (2015) that, because the distribution of envelope masses of the progenitors is non-symmetrical, the duration distribution corresponding to the collapsar scenario might not be symmetrical. Chattopadhyay & Maitra (2017) even found five kinds of GRBs in the BATSE catalog using Gaussian Mixture model. However, some recent works still insisted on two components (Tarnopolski 2015(Tarnopolski , 2016Yang et al. 2016b;Kulkarni & Desai 2017;Bhave et al. 2017), and showed the intermediate GRB class is unlikely. The plausible explanation of the duration bimodal distribution is that the two-class GRB duration distributions are intrinsically non-symmetrical (Tarnopolski 2016). This indicates that we need more GRB parameters to evaluate the optimum number of components, and T 90 should not be the unique criterion. Because we have found some SGRBs have the properties of LGRBs, and LGRBs have the properties of SGRBs, duration criterion is not enough to reveal the physical origin of the GRBs. For example, GRB 060505, GRB 060614 and GRB 111005A have T 90 greater than 2 s, but we have no detection of supernova associated with these three GRBs (Dong et al. 2017). They are also named as long-short GRBs or SN-less LGRBs (Wang et al. 2017). GRB 060614 is also more like an SGRB from the temporal lag and peak luminosity (Gehrels et al. 2006). Several models were proposed to explain GRB 060614, including the merger of an NS and a massive WD (King et al. 2007), the tidal disruption of a star by an intermediate-mass black hole (Lu et al. 2008), and a stellar-mass BH and a WD (Dong et al. 2017). A near-infrared bump was discovered in the afterglow, which probably arose from a Li-Paczyński macronova (Li & Paczyński 1998;Dong et al. 2017), which supports the compact binary merger model for GRB 060614 (Yang et al. 2015). On the contrary, the T 90 of GRB 090426 is 1.24 s, but it lies in a blue, star-forming and interacting host galaxy. The afterglow located at a small offset from the center of its host galaxy, and it is in the LGRB region of E p,rest -E iso plot (Antonelli et al. 2009;, where E p,rest is spectral peak energy in rest-frame, E iso is istropic γ-ray energy. All these properties indicate that its origin might be collapsing of massive star. Zhang et al. (2009a) suggested that we need multi-wavelength criteria to decide the physical origin of individual GRBs.  gave a comparative overlapping properties study of SGRBs and LGRBs, and found that the three best parameters for the classification purpose are T 90 , f eff and F light , where f eff is the effective amplitude parameter ) and F light is surface brightness fraction. Ruffini et al. (2016) divided LGRBs and SGRBs further into two sub-classes, depending on whether a BH is formed in the merger or in the hypercritical accretion process exceeding the NS critical mass, and they indicated two additional progenitor systems: WD-NS and BH-NS. Therefore, with the complex of the GRBs, the muti-dimensional analysis based on more complete data is likely to reveal the true classification, which needs a comprehensive data collection.
The other method is to seeking the underlying correlation between different properties. The most quoted relations are Amati relation (Amati et al. 2002;Sakamoto et al. 2004;Lamb et al. 2004;Amati 2006;Amati et al. 2009;Virgili et al. 2012;Demianski et al. 2017) and Ghirlanda relation (Ghirlanda et al. 2004b). Amati et al. (2002) found a correlation between E iso in the rest-frame 1-10 4 keV energy band and spectral peak energy (E pk ) with 12 BeppoSAX GRBs. It is maybe due to larger initial Lorentz factor Γ 0 in GRBs also have larger energy and peak energy , or an optically thin synchrotron shock model (Lloyd et al. 2000b). Amati et al. (2008) updated the samples. They used 70 LGRBs and XRFs (X-ray flashes), and analyzed the correlation between rest-frame peak energy (E pk,i ) and E iso . They also pointed that E pk,i -E iso correlation is not affected by significant selection effects. However, this relation is also challenged by other works (Nakar & Piran 2005;Band & Preece 2005). Ghirlanda et al. (2004b) used 40 GRBs to derive a correlation between collimation-corrected energy and E pk . They gave a further test for this correlation later . Yonetoku et al. (2004) also gave a reliable relation between E pk and peak luminosity L pk (Yonetoku correlation) using the data of BeppoSAX and BATSE.

5
Then they estimated redshifts and GRB formation rate for some GRBs without known distances. Afterwards, a growing number of samples are used to investigate Yonetoku correlation (Ghirlanda et al. 2004a(Ghirlanda et al. , 2005Yonetoku et al. 2010;Lu et al. 2010;Tsutsui et al. 2013b). Yonetoku et al. (2010) reanalyzed Amati and Yonetoku relation with 101 GRBs, and examined how the truncation of the detector sensitivity affects the correlations, and concluded they are surely intrinsic properties of GRBs. Firmani et al. (2006) reported that adding T R45  can reduce the dispersion of Yonetoku correlation. Therefore, they discovered a correlation between L pk , E pk and T R45 . However, the studies of Rossi et al. (2008) and Collazzi & Schaefer (2008) did not confirm this relation. Tsutsui et al. (2009) investigated the correlation between the residuals of L pk and E iso from the best function, and found that the luminosity time (T L = E iso /L pk ) can improve the Amati and Yonetoku relations. Later, Tsutsui et al. (2011) gave a new correlation between T L , L pk and E pk with considering the systematic errors. Tsutsui et al. (2010) pointed that the intrinsic dispersion of correlations among E pk , L pk and E iso depends on the quality of data set. Tsutsui et al. (2013b) analysed Amati and Yonetoku relations for 13 SGRBs, and the correlations are dimmer than those of LGRBs for the same E pk . Zhang & Wang (2017) used the E pk and L pk correlation with 16 SGRBs to study the luminosity function and formation rate of SGRBs. Isotropic luminosity (L iso ) and E pk are also found to have good correlation (Schaefer 2003a;Frontera et al. 2012;Nava et al. 2012). Schaefer et al. (2001) and Schaefer (2003a) pointed that the E pk and L iso correlation is due to their dependence on Γ 0 . With this correlation, we can further study the structure of the ultra relativistic outflow, the shock acceleration and the magnetic field generation (Lloyd-Ronning & Petrosian 2002). Dichiara et al. (2016) found a highly significant anticorrelation between E pk,i and power density spectra (PDS) slope α from 123 LGRBs. They put forward a model based on magnetic reconnection for this phenomenon. The E pk,i and α are linked to the ejecta magnetization at the dissipation site, so that more magnetised outflows would produce more variable GRB light curves at short timescales, shallower PDS and higher values of E pk,i . Addition to the relations extracted from the spectra, some relations are obtained from the GRB light curves. Norris et al. (2000) found an anti-correlation between L pk and spectral lag with 174 BATSE GRBs. Tsutsui et al. (2008) got a new spectral lag-L pk relation with 565 BATSE GRBs, which is different with the result of Norris et al. (2000). This anti-correlation might contain indirect connections to Γ 0 , and it has been confirmed by several studies (Salmonson 2000;Schaefer et al. 2001;Daigne & Mochkovitch 2003;Zhang et al. 2006b). The interpretation of this relation might be kinematic effect (Salmonson 2000), or energy formation affecting the development of the pulse much more than dissipation (Norris et al. 2000). The correlation between variability and L pk can estimate GRB redshift and luminosities (Fenimore & Ramirez-Ruiz 2000; Reichart et al. 2001). The origin of this correlation may be relativistically shocked jets (Schaefer 2007). Mallozzi et al. (1995) found a correlation between E pk and peak photon flux (P pk ) in 256 ms time bin of 50-300 keV. Then Lloyd et al. (2000a) simulated 1000 GRBs in 50-300 keV energy band, and found a similar strong correlation between E pk and F g . Goldstein et al. (2010) used this correlation to classify LGRBs and SGRBs, and confirmed the presence of two GRB classes. There are many similar works to confirm this correlation (Borgonovo & Ryde 2001;Ghirlanda et al. 2010;Guiriec et al. 2010;Ghirlanda et al. 2011;Lu et al. 2012a). With the estimated Γ 0 , some related correlations are also found, such as Γ 0 − E iso (Liang et al. 2010), Γ 0 − L iso ) and L iso − E pk,i − Γ 0 (Liang et al. 2015). Willingale et al. (2007a) showed that a source frame characteristic-photon-energy/peak luminosity ratio, K z , can be constructed, and it is constant within a factor of 2 for all bursts. The existence of K z indicates that the mechanism responsible for the prompt emission from all GRBs is probably predominantly thermal (Willingale et al. 2007a).  analyzed the individual prompt pulses of a GRB. They showed that the luminosity of pulses is correlated with the peak energy of the pulse spectrum, and anti-correlated with the time since ejection of the pulse.
Besides the individual study on GRB prompt emission, the statistics on the afterglows is also helpful in understanding the nature. A correlation was found between the early optical/UV luminosity (measured at rest-frame 200 s) and average decay rate (measured from 200 s) (Oates et al. 2009(Oates et al. , 2012. The luminosity-decay correlation also exits in X-ray band, and is consistent with optical/UV (Oates et al. 2015(Oates et al. , 2016Racusin et al. 2016). At the same time, the early optical/UV luminosity (measured at rest-frame 200 s) correlations with isotropic energy E iso and rest-frame peak spectral energy E pk (Oates et al. 2015;D'Avanzo et al. 2012;Margutti et al. 2013). Evans et al. (2007) and Evans et al. (2009) provided the methods and results of an automatic analysis of a complete sample of Swift-XRT observations of GRBs. Liang et al. (2008b) and Liang et al. (2008a) gave more detailed analysis, like the jet breaks (Liang et al. 2008a), and the early shallow decay to late jet like decay phases (Liang et al. 2008b). Yi et al. (2016) gave a comprehensive study of the X-ray flares from GRBs observed by Swift. They analyzed the ten-year X-ray flare data of Swift/XRT until the end of March 2015, and studied the distributions of energy, duration, waiting time, rise time, decay time, peak time and peak flux. After that, Yi et al. (2017) statistically studied GRB optical flares from the Swift/UVOT catalog. They found optical flares and X-ray flares may share the similar physical origin and both of them are possibly related to central engine activities. Jia et al. (2016) gave a statistical study of GRB X-ray flares to prove the ubiquitous bulk acceleration in the emission region. With the recent development of networks of robotic telescopes, we are able to follow up the early optical emission of GRBs and to find a peak in the optical afterglow light curves (t pkOpt ) of some GRBs. It is due to the dynamics of the fireball deceleration. So t pkOpt can provide the Γ 0 of the fireball before deceleration, and Γ 0 represents the maximum value attained by the outflow during this dynamical evolution. Bulk Lorentz factor Γ 0 of GRBs is very important. We can use it to compute GRBs' comoving frame properties shedding light on their physics . We can use t pkOpt to derive the distribution of Γ 0 (Rykoff et al. 2009;Ghirlanda et al. 2012Ghirlanda et al. , 2017, also other methods to estimate Γ 0 (Sari & Piran 1999;Molinari et al. 2007;Zou & Piran 2010;Ghisellini et al. 2010;Zou et al. 2011;Nava et al. 2013;Nappo et al. 2014;Zou et al. 2015). Geng & Huang (2016) developed a numerical method to calculate the dynamic of the system consisting of a forward shock and a reverse shock. They found that the steep optical re-brightenings would be caused by the fall-back accretion of black holes, while the shallow optical re-brightenings are the consequence of the injection of the electron-positron-pair wind from the central magnetar. Dainotti et al. (2008) examined the X-ray decay curves of all the GRBs measured by the Swift that available. They found a correlation between log[T a /(1 + z)] ( the time in the X-ray at the end of the plateau in rest-frame) and log[L X (T a )] (X-ray luminosity at the time T a ), hereafter also referred as LT. The slope is −0.74 +0.2 −0.19 . This LT anti-relation shows that the shorter the plateau duration, the more luminous the plateau. Dainotti et al. (2008) believed that this LT anti-correlation is a further tool towards the standardization of GRBs as a distance indicator. This result is confirmed by Ghisellini et al. (2009) andYamazaki (2009). A physical subsample of LGRBs with a significant LT anti-correlation in the GRB rest-frame is discovered (Dainotti et al. 2010). Dainotti et al. (2016) revealed that the subsample of LGRBs associated with SNe presents a very high LT anti-relation. This analysis may open new perspectives in future theoretical investigations of the GRBs with plateau emission and associated with SNe (Dainotti & Del Vecchio 2017). Many researches have been carried on the LT relation, like expanding the sample (Dainotti et al. 2011a;Mangano et al. 2012;Dainotti et al. 2015b), or the selection biases influence on the slope of the relation (Dainotti et al. 2013a;Dainotti & Amati 2018;).There are also many correlations between the prompt and the afterglow. For the LT relation, Xu & Huang (2012) added E iso as a third parameter to get a tighter three-parameter correlation. When adding peak luminosity in the prompt emission L pk , Dainotti et al. (2016) also found a good correlation. Similarly, Si et al. (2018) analyzed the optical light curves of 50 GRBs. They calculated the break time of optical plateaus and the break luminosity. When they added E iso , a significantly tighter correlation was found. Dainotti et al. (2015a) demonstrate that L pk (the peak luminosity in the prompt emission) and L prompt (the averaged prompt luminosity) have intrinsic correlations with L X (T a ) respectively. Liang & Zhang (2005) derived a three parameters correlation, which is between E γ,iso , spectral peak energy in the rest-frame and rest-frame break time of the optical afterglow light curves. Zaninoni et al. (2016) gave a comprehensive statistical analysis of Swift X-ray light curves of GRBs collected from December 2004 to June 2014, and found a three parameter correlation between E iso in the rest-frame 1-10 4 keV energy band, E pk,i and E X,iso (X-ray energy emitted in the rest frame 0.3-30 keV) (Margutti et al. 2013).
There are also some researches on GRB host galaxies. The LGRBs host galaxy offsets (the distance from the site of the GRBs to the center of its host galaxies) are consistent with the expected distribution of massive stars , and SGRBs host galaxy offsets are in good agreement with neutron star binary mergers (Fong et al. 2010a;Church et al. 2011). SGRBsEE seems to be the subgroup of SGRBs, because SGRBsEE mostly have smaller projected physical offsets (Troja et al. 2008) and occur closer to their host galaxies in denser interstellar environments (Malesani et al. 2007). SGRBs without extended emission are opposite. It also implies that SGRBs possibly have two distinct populations. SGRBsEE are due to NS-BH mergers, and SGRBs without extended emission are due to NS-NS mergers (Troja et al. 2008). Furthermore, X-ray absorption column densities and SGRBs host galaxy offsets correlation gives another evidence that SGRBs possibly have two distinct populations . Japelj et al. (2016) studied the host galaxies of a complete sample of bright LGRBs to investigate the impact of the environment on GRB formation. Arabsalmani et al. (2018) studied the mass-metallicity (MZ) relation from 33 GRB hosts spanning a redshift range between ∼ 0.3 and ∼ 3.4 and a mass range from 10 8.2 M to 10 11.1 M . They found that GRB hosts track the the MZ relation of the general star-forming galaxy population with Statistics on GRBs 7 an average offset of 0.15±0.15 dex below the MZ relation of the general population, and metallicity measurements can influence the relation result (Arabsalmani et al. 2018). The offset may be the result of the different methods used to select their respective galaxy populations (Kocevski & West 2011). There are also some relations between host galaxy and afterglow emission (or prompt emission). Zhang et al. (2017) carried out some statical analysis and found possible correlations between SGRBs afterglow luminosities and their host galaxy offset. This may be due to the number density of circum-burst medium should decrease with the distance to the host galaxy center. However there are some other uncertainties related to the correlation, including the angle between the line of sight, the host galaxy disk and SGRBs occurred in globular cluster (Zhang et al. 2017). Selection effects usually occur when the sample observed is not representative of the true population itself (Dainotti & Del Vecchio 2017).
All the previous statistical studies are all based on the data directly from the original data or the catalogs. For each individual study, one needs to collect the data from the beginning. Here we are trying to collect all the data and let the data as a reservoir for future studies. The data are all manually collected from the literature, including almost all the properties belongs to GRBs, i.e., prompt emission, afterglows and the host galaxies. Comparing with the automatic gathering data by machine, the manual collection is slow. However, it is hard to let the machine to recognize the variety of symbols, and many of data are expressed in different ways, which makes the task even harder. Before the advance of the machine algorithm, for the precision of the data gathering, we can only manually collect them in the present stage. This paper is organized as follows. In Section 2, we introduce the data preparation, including data gathering from published papers. The table is in a machine readable table, while a sample of the table is shown in Table 1. Error imputation for the data with central values but without error bars is introduced in Section 3. In Section 4, we introduce our statistical methods. We give the distributions for each parameter in observer-frame and some parameters in rest-frame. All the histograms are shown in Fig. Set 1. We put two histograms in Figure 1. We gave scatter plots between two parameters with at least 5 samples. All the plots are shown in Fig. Set 2 and two scatter plots in Figure 2. We analyzed the linear coefficient and non-linear correlation ratio between two parameters with at least 5 GRBs. We excluded the trivial results, like T 90 and T 90,i correlation, which are obviously correlated but do not give extra information. A small part of all the results are shown in Table  3, and the comprehensive results are shown in machine the readable table. We gave all the linear regression results between two and three parameters with at least 5 GRBs. A small part of all the results are shown in Table 4 and Table 5, and all the results are given in two machine readable tables. Conclusions are given in Section 5 and Section 6, we gave more detailed analysis for some good results, and gave some reasonable explanation for these good results. Discussion is given in Section 7 and conclusion is given in Section 8. A concordance cosmology with parameters H 0 = 67.8 ± 0.9km s −1 Mpc −1 , Ω M = 0.308 ± 0.012 (Planck Collaboration et al. 2016) is adopted in all part of this work.

SAMPLES
We collected all the possible data for 6289 GRBs from GRB 910421 (April 21st, 1994) to GRB 160509A (May 9th, 2016). There are 46 parameters in this catalog, which includes the basic information, the prompt emission, the afterglow and the host galaxy.
For basic information of each GRB, we recorded the trigger time, instrument, trigger number, coordinate and position error. Most of basic information is from Gamma-ray Coordinates Network (GCN) and published papers, while most of BATSE GRBs information is from BATSE website 1 . The basic information is quite important. It can help us to make sure weather some GRBs with the same names are different GRBs or not. Because for some GRBs detected by different instruments have the same GRB name, we changed the names for a small part of GRBs. For example, there is a Fermi GRB 100911, while there is also a MAXI GRB 100911. They have same name in different papers, however, the trigger time is different. Therefore, they must be different GRBs. We made the Fermi GRB name as GRB 100911A and the MAXI GRB name as GRB 100911B, following the trigger time. Besides, some GRBs have different names in different papers. For all the GRBs that have different names or are changed the names by us, we use the flag "A" to remark. Not all the basic information are available for every GRB. When we did not find the information, we let the place empty.
The prompt emission properties include z (redshift), D L (luminosity distance, 10 28 cm), T 90 , T 50 , T R45 (defined in Reichart et al. 2001), variability 1 (variability of Fenimore & Ramirez-Ruiz (2000) definition), variability 2 (variability of Reichart et al. (2001) definition), variability 3 (variability of Schaefer (2007) definition), F g (fluence in 20-2000 keV energy band, in unit of 10 −6 ergs cm −2 ), HR (hardness ratio between 100-2000 keV and 20-100 keV),E iso (isotropic γ-ray energy in rest-frame 1-10 4 keV energy band, in unit of 10 52 ergs), L pk (peak luminosity of 1 s time bin in rest-frame 1-10 4 keV energy band, in unit of 10 52 erg s −1 ), F pk1 (peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, in unit of 10 −6 ergs cm −2 s −1 ), F pk2 (peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, in unit of 10 −6 ergs cm −2 s −1 ), F pk3 (peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, in unit of 10 −6 ergs cm −2 s −1 ), F pk4 (peak energy flux of 1024 ms time bin in rest-frame 1-10 4 keV energy band, in unit of 10 −6 ergs cm −2 s −1 ), P pk1 (peak photon flux of 64 ms time bin in 10-1000 keV, in unit of photons cm −2 s −1 ), P pk2 (peak photon flux of 256 ms time bin in 10-1000 keV, in unit of photons cm −2 s −1 ), P pk3 (peak photon flux of 1024 ms time bin in 10-1000 keV, in unit of photons cm −2 s −1 ), P pk4 (peak photon flux of 1 s time bin in 10-1000 keV, in unit of photons cm −2 s −1 ). α band (low energy spectral index of band model), β band (high energy spectral index of band model), E p,band (spectral peak energy of band model, keV), α cpl (low energy spectral index of CPL model), E p,cpl (spectral peak energy of CPL model, in unit of keV), α spl (spectral index of SPL model), θ j (jet open angle, in unit of rad), spectral time lag (in unit of ms MeV −1 ), Γ 0 (initial Lorentz factor). We removed the spectral time lag for GRB 060218, because it is an clear outlier (Foley et al. 2008). P pk and F pk have four time bins: 64 ms, 256 ms, 1024 ms and 1 s. We make them as different parameters. As the data are from different instruments, while different instruments have different energy bands and different sensitivities. In order to avoid the influence of different energy band, we corrected the P pk into observer-frame in 10-1000 keV energy band, F pk into rest-frame 1-10 4 keV energy band using the method of Schaefer (2007) of four time bins respectively. At the same time, using the same method, we changed all the F g into 20-2000 keV, HR into 100-2000/20-100 keV, E iso into rest-frame 1-10 4 keV energy band, L pk into rest-frame 1-10 4 keV energy band. Besides, we calculated a small part of E iso also using the method of Schaefer (2007) with F g and z. We gave different flags for different data calculation and correction.
The afterglow properties include log t burst (central engine active duration in logarithm, in unit of s) (defined in Zhang et al. 2014a), t pkX (peak time in X-ray LC, in unit of s), t pkOpt (peak time in optical LC, s), F X11hr (flux density in X-ray band at 11 hours related to the trigger time, in unit of Jy), β X11hr (index in X-ray band at 11 hours related to the trigger time), F Opt11hr (flux density in optical band at 11 hours after the trigger time, Jy), t radio,pk (peak time in radio band, in unit of s), F radio,pk (peak flux density in radio band at 8.46 GHz, in unit of Jy).
The host galaxy properties include host galaxy offset (the distance from GRB location to the centre of its host galaxy, in unit of kpc), metallicity (12 + log [O/H]), Mag (absolute magnitude in AB system at rest 3.6 µm wavelength), N H (column density of hydrogen, in unit of 10 21 cm −2 ), A V (dust extinction), SFR (star formation rate, in unit of M yr −1 ), log SSF R (specific star formation rate in logarithm, in unit of Gyr −1 ), Age ( in unit of Myr), log M ass (stellar mass in logarithm, in unit of M ). For metallicity, we converted different metallicity calibrations into Kobulnicky & Kewley (2004) calibrator using the method given in Kewley & Ellison (2008).
We also changed 10 parameters from observer-frame to rest-frame, including T 90 , T 50 , T R45 , E p,band , E p,cpl , spectral time lag, log t burst , t pkX , t pkOpt and t radio,pk . We use label "i" to remark the parameters in rest-frame. We put the 10 rest-frame parameters as new parameters joining the statistics. But we did not show the results for the same parameter just in different frames, such as T 90 and T 90,i . In the table, we also did not include the 10 restframe parameters as they can be easily obtained. When we collected the data from the literatures, some data have different units for the same parameter. We convert them into the same. For example, the unit of D L (luminosity distance) includes 10 28 cm, kpc and others. In order to have an easy conversion in ergs for E iso calculation, we choose the unit 10 28 cm for D L . The units of the parameters are all given in table 1.
All the information above of the 6289 GRBs are collected in the machine readable table. We just put the first 5 samples in Table 1 as an example. In the machine readable table, every GRB has the basic information and 46 parameter values. If the basic information is not available, we let it as blank. If the parameter value is not available, we use "..." to remark. For every available parameter value, we also put the relevant reference or flag (or both). For different situations, we use different flags. We introduce every flag one by one in the following. "a" means the errors are imputed by MICE algorithm. The details of error imputation are shown in Section 3. "b" means the errors in the original papers were in 90% confidence level, and we change the errors to 1 σ confidence level by multiplying 0.995/1.645. "c" means the values are calculated using the spectral values in order to change the energy band to a uniform band. Because for some parameters, the energy band is different for different instruments.

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In order to avoid the influence of different energy band, we use the spectral values to correct the energy band of the 4 time bins P pk , 4 time bins F pk , F g , HR, E iso and L pk . "d" means the unit is different from the original papers. For example, the unit for D L in Cano et al. (2014) is Mpc, we change it to 10 28 cm. "e" means the error is estimated as the central value multiplying 0.1. One can see Section 3 for more details. When coming to spectral values, sometimes the spectral index cannot be constrained very well. We use the BASTE α peak value -1.1 as common with flag "f", and the BASTE β peak value -2.2 as common with flag "g". Some values are in rest-frame, then we change the values from rest-frame to observer-frame with flag "h". "i" means the values are converted into logarithm or from logarithm into the normal form. "j" means the values are calculated using other parameters. For example, almost half of HRs are calculated from F g and spectral values. "k" means we converted different metallicity calibrations into Kobulnicky & Kewley (2004) calibrator using the method in Kewley & Ellison (2008). We found the metallicity is different with different calibrators, so this step is necessary. "m" means the D L is calculated using z with cosmology parameters in Planck Collaboration et al. (2016). There is also a description in Table 1.
The data are from different literatures, GCN, websites 2 and calculations if it is not directly available and can be derived. Principle of data collection is the following. We collect variables as many as possible. We only take the certain values, which are mainly taken from literature manually. Every data should be directed reversely to the original reference. The data should include error bars in 1 σ. If the error is in other confidence, we convert the error into 1 σ. If it is not available, we use the imputation, which is shown in Section 3. For those data are shown in different literature, we choose the values in the following order (higher to lower priority): 1st, published papers (catalogs, data gatherings, other articles); 2nd, GCNs; 3rd, other websites; 4th self-calculation if they are able to obtain. For those data shown in ordinary published papers, the paper from the official team is in higher priority. Otherwise, the newer paper is of higher priority.
We excluded Fermi GRBs 120222A, 110920A, 110517A and 101214A, because they have the same GRB names while two different trigger numbers . This might be a mistake for GRB names. For GRB 091024A, it triggered Fermi /GBM twice and triggered Swift once ). We set it as one GRB for the analysis. During the statistical study, we deleted some values with central values smaller than error bars (which means the values in the lower side are not physical). Because for some parameters, the lower limit must be greater than 0, such as T 90 . If the central value is smaller than error bar for T 90 , the lower limit is smaller than 0. For example, in von Kienlin et al. (2014), the T 90 of GRB 080719A is 16.128 ± 17.887 s, we did not include this datum. Notice we still put this kind of data in the table, and just did not include them in the statistics.
For the spectral parameters, the spectra are mainly fitted by three models: Band model, cutoff power law (CPL) model and simple power law (SPL) model ). Band model is a smoothly joint broken power law with the definition (Band et al. 1993): where α is low energy photon index, β is high energy photon index, A is the coefficient for normalization, E is the energy of the photons, and E 0 is the break energy. Mostly we used E p instead of E 0 . E p is the peak energy in spectrum E 2 N , and E p = (2 + α)E 0 . In the big table, we use α band , β band and E p,band as Band function spectral parameters. α cpl and E p,cpl to mark CPL model spectrum parameters. The formula of SPL model is Most of the data are taken from the following literatures: (1) Swift lookup website 3 with the number of selected data 1003; (2) "How long does a burst burst?" ) with the number of selected data 354; (3) "Cosmic evolution of long gamma-ray burst luminosity" (Deng et al. 2016) with the number of selected data 177. (4) "Effect of gamma-ray burst (GRB) spectra on the empirical luminosity correlations and the GRB Hubble diagram" (Lin et al. 2016) with the number of selected data 219; (5) "On the classification of GRBs and their occurrence rates" (Ruffini et al. 2016) with the number of selected data 571; (6) "The third Fermi GBM gamma-ray burst catalog: the first six years"  with the number of selected data 3702; (7) "A comparative study of long and short GRBs. I. overlapping properties"  with the number of selected data 786; (8) "GRB hosts through cosmic time. VLT/X-Shooter emission-line spectroscopy of 96 γ-ray-burst-selected galaxies at 0.1 < z < 3.6"  with the number of selected data 107; (9) "Uncovering the intrinsic variability of gamma-ray bursts" (Golkhou & Butler 2014) with the number of selected data 323; (10) "The E peak -E iso relation revisited with Fermi GRBs. Resolving a long-standing debate" (Heussaff et al. 2013) with the number of selected data 671; (11) "The Second Fermi GBM Gamma-Ray Burst Catalog: The First Four Years" (von Kienlin et al. 2014) with the number of selected data 2019; (12) "The spectral catalogue of INTEGRAL gamma-ray bursts. results of the joint IBIS/SPI spectral analysis" ) with the number of selected data 138; (13) "The Swift GRB Host Galaxy Legacy Survey. II. Rest-frame Near-IR Luminosity Distribution and Evidence for a Near-solar Metallicity Threshold" ) with the number of selected data 132; (14) "The dark bursts population in a complete sample of bright Swift long gamma-ray bursts" ) with the number of selected data 107; (15) "A Radio-selected Sample of Gamma-Ray Burst Afterglows" (Chandra & Frail 2012) with the number of selected data 347; (16) "Statistical Analysis of the Observable Data of Gamma-Ray Bursts"  with the number of selected data 260; (17) "Spectral properties of 438 GRBs detected by Fermi/GBM" ) with the number of selected data 694; (18) "Possible Origins of Dispersion of the Peak Energy-Brightness Correlations of Gamma-Ray Bursts" (Yonetoku et al. 2010) with the number of selected data 254; (19) "The Cosmic Rate, Luminosity Function and Intrinsic Correlations of Long Gamma-Ray Bursts" ) with the number of selected data 874; (20) "The updated spectral catalogue of INTEGRAL gamma-ray bursts" (Vianello et al. 2009) with the number of selected data 121; (21) "Redshift Catalog for Swift Long Gamma-ray Bursts" (Xiao & Schaefer 2011) with the number of selected data 415; (22) "Search for gamma-ray burst classes with the RHESSI satellite" (Řípa et al. 2009) with the number of selected data 323; (23) "Statistical studies of optically dark gamma-ray bursts in the Swift era" ) with the number of selected data 271; (24) "Correlations of Prompt and Afterglow Emission in Swift Long and Short Gamma-Ray Bursts" (Gehrels et al. 2008) with the number of selected data 209; (25) "A Complete Catalog of Swift Gamma-Ray Burst Spectra and Durations: Demise of a Physical Origin for Pre-Swift High-Energy Correlations" (Butler et al. 2007) with the number of selected data 1069; (26) "Intrinsic properties of a complete sample of HETE-2 gamma-ray bursts. A measure of the GRB rate in the Local Universe" (Pélangeon et al. 2008) with the number of selected data 272; (27) "Global characteristics of X-ray flashes and X-ray rich GRBs observed by HETE-2" (Sakamoto et al. 2005) with the number of selected data 101; (28) "Konus catalog of SGRBs"  with the number of selected data 128; (29) "The Gamma-Ray Burst Catalog Obtained with the Gamma-Ray Burst Monitor Aboard BeppoSAX" (Frontera et al. 2009) with the number of selected data 1619; (30) "Spectral catalogue of bright gamma-ray bursts detected with the BeppoSAX/GRBM" (Guidorzi et al. 2011) with the number of selected data 248; (31) "BASTE current gamma-ray burst catalog 4 with the number of selected data 8983; (32) "Hardness as a spectral peak estimator for gamma-ray bursts" (Shahmoradi & Nemiroff 2010) with the number of selected data 2128; (33) "The BATSE 5B Gamma-Ray Burst Spectral Catalog"  with the number of selected data 4764; (34) "Are There Any Redshift > 8 Gamma-Ray Bursts in the BATSE Catalog" (Ashcraft & Schaefer 2007) with the number of selected data 110; (35) "Short versus long gamma-ray bursts: spectra, energetics and luminosities" ) with the number of selected data 139; (36) "The Fourth BATSE Gamma-Ray Burst Catalog (Revised)" (Paciesas et al. 1999) with the number of selected data 1049; (37) "On the Spectral Energy Dependence of Gamma-Ray Burst Variability" (Lloyd-Ronning & Ramirez-Ruiz 2002) with the number of selected data 159; (38) "The Third BATSE Gamma-Ray Burst Catalog"  with the number of selected data 1380. A full reference list is given following Table 1.

ERROR IMPUTATION
In the table, there are many incomplete data. Especially there are many data just having central values without error bars. However, when we perform the statistics, we need to consider the uncertainty for each data. Therefore, we used the R package mice imputes incomplete multivariate data by multiple imputation (Rubin 1987(Rubin , 1996 by chained equations (MICE). Multiple imputation is the method to deal with the problems of incomplete data. In the big table, many parameters have missing errors. For the first step of generating multiple imputation, we used the predictive mean matching (Little 1988) (PMM) method. It is a general purpose semi-parametric imputation method for numeric scale type. We did 5 times iterations for every missing error, and we set a threshold 0.25 for imputation. It means the error of each parameter should have at least 25% percentage information. Because if the information is not enough, the imputation maybe not reliable. Then we use three indictors to asses the imputation results: RIV (relative increased variance), FMI (fraction of missing information) and RE (relative efficiency) (Rubin 1987(Rubin , 1996, One can see the results in Table 2. However, there are some parameters that we cannot impute the missing errors. Then we use the 10% of the absolute central value as the error bars (Schaefer 2007). The errors of t pkX , F Opt11hr , Mag, log SSF R, Age, t pkX,i used the 10% of the absolute central value as the error bars. We do not impute the errors of z and D L , as the errors are relatively very small.
1. Relative increase in variance due to missing data r m (RIV). It is the ratio of between-imputation variance and within-imputation variance of the 5 data sets, then multiplying the imputation time m. It stands for the increase fraction in variance due to missing data, the influence of the missing data is bigger when r m is bigger. When r m is smaller, it indicates influence of the change of m is smaller, this is to say that missing data has smaller influence to the whole data parameters, the imputation results are more stable, the imputations are better. r m is defined as is between-imputation variance, and θ i is the mean of every complete data set, θ = 1 m m i=1 θ i . σ W stands for the mean of the variance of m data sets, and σ B stands for the variance of the mean of m data sets.
2. Fraction of missing information γ m (FMI). It stands for the influence of the missing data for the whole parameters (e.g. mean), FMI is smaller, the imputation results are more stable.
3. Relative efficiency (RE). RE is a comprehensive analysis of RIV and FMI, it stands for the imputation fraction for missing information by MICE. RE is bigger, the results are better.
For analyzing imputed data and pooling analysis results, we use the mean of every imputed error bar, because we also need to calculate some values and plot scatter plots with error bars, we can not use 5 values for one error bar, we must get a best estimate for every imputed error bar, so we use the mean.
The imputation results are shown in Table 2. From the results, we can see that RIV and FMI are very close to 0, which means the imputation is stable. RE is very close to 1, which means the imputation efficiency is very high. We almost imputed all the missing information. Therefore, the imputation is reliable.

Distributions for each parameter
We did the histograms for every parameter in observer-frame and some parameters in rest-frame. Two histograms are shown in Figure 1. We made the figure set for all the figures, which are in total 58 figures. Readers can see all the figures in electronic version.

scatter plots between two arbitrary parameters
We plotted the scatter plots between two arbitrary parameters when the points is not less than 5. We can find that, with this large samples, we can also keep the relationships that found before. We removed the spectral lag value of GRB 060218, because it is an outlier (Foley et al. 2008). The two results are shown in Figure 2. We also made figure set for all the scatter plots. In total there are 1468 plots.
Fig. Set 2. Scatter plots for two arbitrary parameters

Correlation coefficients
Correlation coefficient is to measure the correlation between two parameters, there are 4 methods: 1. Pearson correlation coefficient (Pearson 1895). Supposed the two variables are x t and y t .

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wherex is the mean of variable x,ȳ is the mean of variable y. Pearson correlation coefficient is based on normal distribution, so when the data is normal distribution, the Pearson correlation coefficient works very well.
2. Spearman correlation coefficient (Spearman 1987). This coefficient does not depend on the data distribution. It is a kind of rank measure.
3. Kendall τ correlation coefficient (Kendall 1938). This coefficient compares the order of the two variables, not comparing the values. It is also a kind of rank measure.
where n is the length of the two variables, n c is the number of concordant pairs, n d is the number of discordant pairs.

Cosine similarity (van Dongen & Enright 2012
). Supposed the two variables are x t and y t , the cosine similarity definition is when cosθ x,y = 1, it means x t and y t are completely similar, when cosθ x,y is more close to 1, the two variables are more similar.

Correlation ratio
Pearson correlation coefficients measure the linear correlation between two variables. But when coming to nonlinear correlation, correlation coefficients do not work well. So we need correlation ratio (Fisher 1970) to measure the nonlinear correlation between different variables. Correlation ratio is a ratio of the statistical dispersion within individual categories and the dispersion across the whole population or sample, defined as the ratio of the standard deviation of within individual categories and the standard deviation of the the whole population or sample.
Supposed the variables is y xi , x means the category, i means the ith value for category x, the number of category x is n x ,ȳ whereȳ x is the mean of category x,ȳ is the mean of the whole sample, the definition of correlation ratio η is it is the ratio of the weighted variance of the category means and the variance of the whole sample. η is a value between 0 and 1, when η is more close to 1, the nonlinear correlation is stronger. When the two parameters have at least 5 GRBs, then we gave the correlation coefficients and correlation ratio, and we excluded some trivial results. It means we removed the correlations between the rest-frame and observerframe for the same parameters, like the correlation between T 90 and T 90,i . We have many results, so we just put the first 5 results in Table 3. One can find all the results in machine readable table. In Table 3, we gave the results of Pearson, Spearman, Kendall τ coefficients and the related p-values. We also considered all the error bars using MC method (Zou et al. 2018). The last two rows are correlation ratio and cosine similarity with error bars, also using MC method. We did the statistics between two arbitrary parameters in order to find the remarkable correlations.
In Section 5, we analyzed some interesting results, and gave some reasonable physical explanations. In this paper, we just analyzed the linear correlations. During all the analysis, Pearson, Spearman, Kendall τ coefficients and the related p-values are the most important references of linear correlations. When the hypothesis testing p-value is smaller than 0.1, it means the correlation has very high probability to be true. The sample selection biases is quite complicated (see for examples of the related discussion, (Lloyd & Petrosian 1999;Lloyd et al. 2000a;Dainotti et al. 2013bDainotti et al. , 2015a). One should be caution when considering the results. When the absolute values of Pearson, Spearman, Kendall τ coefficients are bigger, the correlation is stronger. For cosine similarity, there is no hypothesis testing, so it is just used to confirm the linear correlations. For correlation ratio, we will analyzed the nonlinear correlations in the future with this value.

Linear regression of two parameters and three parameters
We did the linear regression between two parameters and three parameters arbitrarily when the sample number is not smaller than 5. We also considered all the error bars using MC method (Zou et al. 2018). We excluded some trivial results. As an example, the first 5 linear regression results are shown in Table 4 and Table 5, respectively. The full linear regression results are given in two machine readable tables.
In Table 4, y is dependent variable, x is independent variable, b is the intercept of the linear model, and a is the linear coefficient of x. Adjusted R 2 is used to measure the goodness of the regression model. It means the percentage of variance explained considering the parameter freedom. We use the adjusted R 2 calculated with central values for representation. Because every time of MC, adjusted R 2 does not change too much. In Table 5, x 1 and x 2 are independent variables, a 1 and a 2 are the linear coefficients of x 1 and x 2 respectively. y, b and adjusted R 2 are same meanings as in Table 4. Readers can see Hron et al. (2012) for more details about linear regression.

REMARKABLE RESULTS BETWEEN TWO PARAMETERS
In this section, we show some remarkable correlations between two parameters. The strategy of choosing the results are the following. The correlation should have at least 10 GRBs. Adjusted R 2 should be bigger than 0.2 except Section 5.1. The whole linear model and a have hypothesis testing p-values smaller than 0.05. Pearson, Spearman, Kendall τ coefficients have at least one hypothesis testing p-value smaller than 0.1. For peak energy flux F pk and peak photon flux P pk , both of them have four different time bins. If F pk in all the four time bins correlates with another one same parameter, we just showed the best time bin result. The best time bin result has biggest adjusted R 2 . P pk is the same. Other results are all in machine readable tables. Since some relations have been discovered before, we compared the differences between the current and previous analysis. Readers can also see Table 6 for a clear comparison. We just show a part of results in the paper. We provide all the results with figure sets and machine readable tables including our original data. If readers want to analyze with other statistical methods, they can use our original data in machine readable table. One may notice that not only the correlations, but the non-existence of correlations among a certain quantities may also reveal underlying physics of GRBs.
We also found there is a break in log HR-log F pk2 -(−α spl ) plot. In other three time bins of F pk , there are also breaks. We just show the best one.
The log HR-log F pk2 -(−α spl ) relation is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.57. One can see 85 for the scatter plot between these three parameters. The GRB sample number is 490. It is not clear what causes these breaks. There is also weak signals in P pk − α spl plot. It might be a selection effect, as there is no such effect in L pk − α spl plot or in F g − α spl plot. All the corresponding figures can be found in the figure sets.

Amati relation
Amati relation (Amati et al. 2002(Amati et al. , 2008) is a widely known relation of GRBs. The correlation between log E p,band and log E iso is: where E p,band is in unit of keV, E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. In Amati et al. (2002) initial data, the slope is 0.52 ± 0.06 with 12 GRBs. Here we have 180 GRBs. The adjusted R 2 is 0.26. The Pearson coefficient is 0.5 ± 0.021 with p-value 8.2 × 10 −13 . The Spearman coefficient is 0.56 ± 0.019 with p-value 5.2 × 10 −16 . The Kendall τ coefficient is 0.41 ± 0.014 with p-value 2.1 × 10 −16 . The correlation ratio is 0.62 ± 0.0049. The cosine similarity is 0.75 ± 0.0055. It is maybe due to larger Γ 0 in GRBs also have larger energy and peak energy , or an optically thin synchrotron shock model (Lloyd et al. 2000b). One can see Section 5.6, E iso and peak energy both correlate with Γ 0 . The scatter plot is in Figure 7. The GRB sample number is 180. The correlation between log E p,band,i and log E iso is: In Amati et al. (2008), the slope is 0.57 ± 0.01 with 70 LGRBs and X-ray flares. We have 178 GRBs. where E p,band,i is in unit of keV. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.45. The Pearson coefficient is 0.65 ± 0.019 with p-value 7.2 × 10 −23 . The Spearman coefficient is 0.69 ± 0.018 with p-value 4.9 × 10 −26 . The Kendall τ coefficient is 0.51 ± 0.015 with p-value 3.1 × 10 −24 . The correlation ratio is 0.72 ± 0.0038. The cosine similarity is 0.78 ± 0.0052. The scatter plot is in Figure 8. The GRB sample number is 178. The better correlation between log E p,band,i and log E iso comparing with log E p,band and log E iso suggests this correlation is intrinsic. With enough data, one would investigate this relation within subgroups (Zhang et al. 2009b;Qin & Chen 2013;Zou et al. 2018 Yonetoku et al. (2004) found the correlation between rest-frame peak energy and peak luminosity. We also did the correlations of log E p,band -log L pk and log E p,band,i -log L pk .
The correlation between log L pk and log E p,band is: where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. E p,band is in unit of keV. The adjusted R 2 is 0.25. The Pearson coefficient is 0.49 ± 0.022 with p-value 3.2 × 10 −9 . The Spearman coefficient is 0.52 ± 0.025 with p-value 1.5 × 10 −10 . The Kendall τ coefficient is 0.37 ± 0.019 with p-value 4.9 × 10 −10 . The correlation ratio is 0.8 ± 0.004. The cosine similarity is 0.33 ± 0.014. The scatter plot is in Figure 10. The GRB sample number is 130. The correlation between log L pk and log E p,band,i is: In Yonetoku et al. (2004), the slope is 1.94 ± 0.19 with 11 samples. where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. E p,band,i is in unit of keV. The adjusted R 2 is 0.42. The Pearson coefficient is 0.63 ± 0.019 with p-value 4 × 10 −15 . The Spearman coefficient is 0.64 ± 0.025 with p-value 4.3 × 10 −16 . The Kendall τ coefficient is 0.47 ± 0.019 with p-value 7.5 × 10 −15 . The correlation ratio is 0.85 ± 0.0031. The cosine similarity is 0.34 ± 0.013. The scatter plot is in Figure 9. The GRB sample number is 127. The rest frame peak energy is also better correlated with the luminosity. Yonetoku et al. (2010)  The correlation between log L pk and log of f set is: where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. Host galaxy offset is in unit of kpc. The adjusted The correlation ratio is 0.42 ± 0.043. The cosine similarity is −0.5 ± 0.085. The scatter plot is in Figure 11. The GRB sample number is 27. As the short GRBs locate in the outer regions of the host galaxies, and they are less luminous, this may cause the anti-correlation between L pk and the offset. The correlation between log L pk and log t pkOpt is: where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. t pkOpt is in unit of s. The adjusted R 2 is 0.26. The Pearson coefficient is −0.51 ± 0.023 with p-value 1.5 × 10 −6 . The Spearman coefficient is −0.39 ± 0.033 with p-value 3.5 × 10 −4 . The Kendall τ coefficient is −0.27 ± 0.024 with p-value 4 × 10 −4 . The correlation ratio is 0.84 ± 0.0038. The cosine similarity is −0.067 ± 0.026. The scatter plot is in Figure 13. The GRB sample number is 80.
The correlation between log L pk and log t pkOpt,i is: where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. t pkOpt,i is in unit of s. The adjusted R 2 is 0.4. The Pearson coefficient is −0.62 ± 0.019 with p-value 1.1 × 10 −9 . The Spearman coefficient is −0.48 ± 0.031 with p-value 5.6 × 10 −6 . The Kendall τ coefficient is −0.33 ± 0.023 with p-value 1.3 × 10 −5 . The correlation ratio is 0.78 ± 0.0049. The cosine similarity is −0.13 ± 0.025. The scatter plot is in Figure 12. The GRB sample number is 80. This anti-correlation is mainly caused by the Lorentz factor, of which higher Lorentz factor corresponds to shorter deceleration time, and consequently smaller t pkOpt,i . We can also see the rest frame peak time is better correlated with the luminosity than the observational peak time.

Some correlations about host galaxy
We found there are some correlations between host galaxy parameters and redshift, also D L . Especially Mag has correlations with many parameters.

Some correlations about Mag and other parameters
The correlation between Mag and log M ass is: where Mass is in unit of M , and Mag is in unit of magnitude. The adjusted R 2 is 0.84. The Pearson coefficient is −0.48 ± 0.077 with p-value 4.2 × 10 −6 . The Spearman coefficient is −0.45 ± 0.083 with p-value 1.8 × 10 −5 . The Kendall τ coefficient is −0.32 ± 0.061 with p-value 2.5 × 10 −5 . The correlation ratio is 0.99 ± 0.00099. The cosine similarity is −0.99 ± 0.00091. The scatter plot is in Figure 14. The GRB sample number is 82. The luminosity of the host galaxy is positively related with its mass. It might be worthy to investigate the correlations for different types of galaxies, such as host galaxies of GRBs, of supernovae, radio load galaxies etc. The correlation between Mag and log SF R is: where SFR is in unit of M yr −1 . Mag is in unit of magnitude. The adjusted R 2 is 0.78. The Pearson coefficient is −0.56 ± 0.1 with p-value 7.1 × 10 −4 . The Spearman coefficient is −0.5 ± 0.12 with p-value 3.1 × 10 −3 . The Kendall τ coefficient is −0.36 ± 0.091 with p-value 3.6 × 10 −3 . The correlation ratio is 0.98 ± 0.0032. The cosine similarity is −0.78 ± 0.027. The scatter plot is in Figure 15. The GRB sample number is 33. It might be more interesting to get the correlations for the specific SFR. However, those correlations are not obvious enough. The correlation between Mag and metallicity is: where Mag is in unit of magnitude. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.41. The Pearson coefficient is −0.42 ± 0.15 with p-value 7.4 × 10 −2 . The Spearman coefficient is −0.41 ± 0.17 with p-value 7.8 × 10 −2 . The Kendall τ coefficient is −0.29 ± 0.13 with p-value 7.4 × 10 −2 . The correlation ratio is 0.99 ± 0.0025. The cosine similarity is −0.99 ± 0.0022. The scatter plot is in Figure 16. The GRB sample number is 19. Metallicity indicates the star formation properties, and consequently the information of the progenitors of GRBs. However, the sample is not big enough, and the correlation is not very tight. One may not find much information from this correlation right now. The correlation between Mag and log F Opt11hr is: where F Opt11hr is in unit of Jy. Mag is in unit of magnitude. The adjusted R 2 is 0.27. The Pearson coefficient is 0.27 ± 0.1 with p-value 3.3 × 10 −2 . The Spearman coefficient is 0.23 ± 0.11 with p-value 7.3 × 10 −2 . The Kendall τ coefficient is 0.16 ± 0.075 with p-value 6.9 × 10 −2 . The correlation ratio is 0.97 ± 0.004. The cosine similarity is 0.98 ± 0.0028. The scatter plot is in Figure 17. The GRB sample number is 64. It is interesting that the magnitude of the host galaxy is related to the observed flux density of the afterglow. It is probably that the number density of the smaller galaxy is higher, and the luminosity at 11 hour is brighter. The correlation between Mag and log D L is: where D L is in unit of 10 28 cm. Mag is in unit of magnitude. The adjusted R 2 is 0.26. The Pearson coefficient is −0.3 ± 0.081 with p-value 6.9 × 10 −3 . The Spearman coefficient is −0.2 ± 0.091 with p-value 7.5 × 10 −2 . The Kendall τ coefficient is −0.14 ± 0.064 with p-value 7.3 × 10 −2 . The correlation ratio is 0.99 ± 0.002. The cosine similarity is −0.82 ± 0.0058. The scatter plot is in Figure 18. The GRB sample number is 81. This relation shows the observational selection effect on the data. Because of the flux limit of the observations, only bright galaxies can be observed in the far distance. The observed host galaxies is correlated to the distance.

Some other correlations between parameters related to host galaxies
The correlation between metallicity and log M ass is: where metallicity is the value of 12 + log O/H. Mass is in unit of M . The adjusted R 2 is 0.3. The Pearson coefficient is 0.47 ± 0.054 with p-value 9.3 × 10 −5 . The Spearman coefficient is 0.49 ± 0.057 with p-value 3.7 × 10 −5 . The Kendall τ coefficient is 0.34 ± 0.043 with p-value 6.7 × 10 −5 . The correlation ratio is 0.54 ± 0.022. The cosine similarity is 1 ± 0.00033. The scatter plot is in Figure 19. The GRB sample number is 64. The metallicity is positively correlated with the mass, which may indicate the history of merging of the galaxies. Arabsalmani et al.
(2018) studied the mass-metallicity relation for GRB host galaxies, and found GRB-selected galaxies appear to track the mass-metallicity relation of star forming galaxies but with an offset of 0.15 towards lower metallicities. The correlation between log SF R and log(1 + z) is: where SFR is in unit of M yr −1 . The adjusted R 2 is 0.49. The Pearson coefficient is 0.69 ± 0.019 with p-value 1.3×10 −14 . The Spearman coefficient is 0.69±0.02 with p-value 4.8×10 −15 . The Kendall τ coefficient is 0.49±0.018 with p-value 1.6 × 10 −12 . The correlation ratio is 0.24 ± 0.018. The cosine similarity is 0.76 ± 0.016. The scatter plot is in Figure 20. The GRB sample number is 96. This relation may indicate the history of the star formation. However, one should be caution about the data selection effect. The correlation between log SF R and log D L is: where SFR is in unit of M yr −1 . D L is in unit of 10 28 cm. The adjusted R 2 is 0.42. The Pearson coefficient is 0.64 ± 0.018 with p-value 3.2 × 10 −12 . The Spearman coefficient is 0.66 ± 0.021 with p-value 2.1 × 10 −13 . The Kendall τ coefficient is 0.46 ± 0.019 with p-value 2.1 × 10 −11 . The correlation ratio is 0.29 ± 0.016. The cosine similarity is 0.69 ± 0.014. The scatter plot is in Figure 21. The GRB sample number is 96. This is similar to the SFR and (1 + z) relation. The correlation between log M ass and log SF R is: where mass is in unit of M . SFR is in unit of M yr −1 . The adjusted R 2 is 0.54. The Pearson coefficient is 0.69 ± 0.034 with p-value 5.2 × 10 −11 . The Spearman coefficient is 0.71 ± 0.034 with p-value 9.6 × 10 −12 . The Kendall τ coefficient is 0.52 ± 0.032 with p-value 2 × 10 −10 . The correlation ratio is 0.98 ± 0.0011. The cosine similarity is 0.57 ± 0.021. The scatter plot is in Figure 22. The GRB sample number is 69. It is natural that the more massive galaxy has more intense total star formation rate. Notice the index is 0.62 ± 0.041, which means the less massive galaxies is more effeminate for the star forming per unit mass.

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The correlation between log F Opt11hr and log SF R is: where F Opt11hr is in unit of Jy. SFR is in unit of M yr −1 . The adjusted R 2 is 0.24. The Pearson coefficient is −0.43 ± 0.065 with p-value 1.7 × 10 −3 . The Spearman coefficient is −0.43 ± 0.069 with p-value 1.6 × 10 −3 . The Kendall τ coefficient is −0.29 ± 0.051 with p-value 2.3 × 10 −3 . The correlation ratio is 0.94 ± 0.0042. The cosine similarity is −0.57 ± 0.03. The scatter plot is in Figure 23. The GRB sample number is 51. This correlation is related to the correlations in eq. (32) and in eq. (27). The correlation between log Age and log E p,cpl is: where E p,cpl is in unit of keV. Age is in unit of Myr. The adjusted R 2 is 0.24. The Pearson coefficient is −0.39 ± 0.1 with p-value 2.2 × 10 −2 . The Spearman coefficient is −0.39 ± 0.11 with p-value 1.9 × 10 −2 . The Kendall τ coefficient is −0.27 ± 0.077 with p-value 2.1 × 10 −2 . The correlation ratio is 0.24 ± 0.049. The cosine similarity is 0.92 ± 0.0086. The scatter plot is in Figure 42. The GRB sample number is 35. The correlation between log Age and log E p,cpl,i is: where Age is in unit of Myr. E p,cpl,i is in unit of keV. The adjusted R 2 is 0. 39 The correlation ratio is 0.069±0.042. The cosine similarity is 0.91±0.0086. The scatter plot is in Figure 43. The GRB sample number is 35. This interesting correlation might be an counter evidence for the two origins of the GRBs. Short GRBs are from merger of double compact objects, which needs longer time of evolution, and consequently in older galaxies. And Short GRBs are harder in spectrum, and the peak energy E p,cpl,i should be higher. Therefore, The age should be positively correlated to the peak energy, while eq. (35) shows it contrarily.

Correlations with HR
The correlation between (−α spl ) and log HR is: the adjusted R 2 is 0.69. The Pearson coefficient is −0.73±0.016 with p-value 1.6×10 −148 . The Spearman coefficient is −0.78 ± 0.011 with p-value 1.9 × 10 −176 . The Kendall τ coefficient is −0.6 ± 0.01 with p-value 8.4 × 10 −158 . The correlation ratio is 0.71 ± 0.0043. The cosine similarity is 0.59 ± 0.0093. The scatter plot is in Figure 24. The GRB sample number is 874. This just shows the consistency of the spectrum. The outliers and the width show the hardness ratio itself is not good enough to present the information of the spectrum. All the correlations below in this section show the similar information.
The correlation between log E p,cpl and log HR is: where E p,cpl is in unit of keV. The adjusted R 2 is 0.65. The Pearson coefficient is 0.69 ± 0.014 with p-value 3.8 × 10 −211 . The Spearman coefficient is 0.78 ± 0.0094 with p-value 6.5 × 10 −305 . The Kendall τ coefficient is 0.61 ± 0.0083 with p-value 1.1 × 10 −267 . The correlation ratio is 0.86 ± 0.0023. The cosine similarity is 0.67 ± 0.008. The scatter plot is in Figure 25. The GRB sample number is 1469. The correlation between log E p,cpl,i and log HR is: log E p,cpl,i = (0.55 ± 0.036) × log HR + (2.6 ± 0.018), where E p,cpl,i is in unit of keV. The adjusted R 2 is 0.6. The Pearson coefficient is 0.63±0.03 with p-value 4.3×10 −27 . The Spearman coefficient is 0.68 ± 0.028 with p-value 1.9 × 10 −32 . The Kendall τ coefficient is 0.49 ± 0.023 with p-value 2.2 × 10 −28 . The correlation ratio is 0.93 ± 0.0032. The cosine similarity is 0.38 ± 0.026. The scatter plot is in Figure 26. The GRB sample number is 229. The correlation between log E p,band and log HR is: where E p,band is in unit of keV. The adjusted R 2 is 0.51. The Pearson coefficient is 0.63 ± 0.017 with p-value 5.2 × 10 −149 . The Spearman coefficient is 0.72 ± 0.01 with p-value 2.3 × 10 −214 . The Kendall τ coefficient is 0.56 ± 0.0081 with p-value 5.9 × 10 −207 . The correlation ratio is 0.9 ± 0.0021. The cosine similarity is 0.81 ± 0.0079. The scatter plot is in Figure 27. The GRB sample number is 1332. The correlation between log E p,band,i and log HR is: where E p,band,i is in unit of keV. The adjusted R 2 is 0.3. The Pearson coefficient is 0.48 ± 0.045 with p-value 4.6×10 −9 . The Spearman coefficient is 0.6±0.035 with p-value 1.4×10 −14 . The Kendall τ coefficient is 0.47±0.028 with p-value 4 × 10 −16 . The correlation ratio is 0.94 ± 0.0025. The cosine similarity is 0.87 ± 0.013. The scatter plot is in Figure 28. The GRB sample number is 136.

Correlations between parameters related with HR or peak energies
The correlation between log F pk1 and log HR is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.26. The Pearson coefficient is 0.45 ± 0.011 with p-value 3.9 × 10 −75 . The Spearman coefficient is 0.47 ± 0.01 with p-value 7.9 × 10 −83 . The Kendall τ coefficient is 0.33 ± 0.0073 with p-value 4.2 × 10 −79 . The correlation ratio is 0.37 ± 0.0044. The cosine similarity is 0.29 ± 0.011. The scatter plot is in Figure 29. The GRB sample number is 1494. Though there is a correlation, one would say it is just a weak tendency.
In Mallozzi et al. (1995), they found a correlation between mean peak energies and 256 ms peak photon flux P pk2 . With accumulated more data, we found P pk2 and peak energy have no obvious correlation with adjusted R 2 smaller than 0.1, except P pk4 and E p,band .
The correlation between log E p,band and log P pk4 is: log E p,band = (0.28 ± 0.03) × log P pk4 + (1.8 ± 0.033), where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV and in unit of photons cm −2 s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.19. The Pearson coefficient is 0.4 ± 0.032 with p-value 1.9 × 10 −8 . The Spearman coefficient is 0.46 ± 0.024 with p-value 4.5 × 10 −11 . The Kendall τ coefficient is 0.31 ± 0.017 with p-value 3.4 × 10 −10 . The correlation ratio is 0.71 ± 0.0087. The cosine similarity is 0.81 ± 0.017. The scatter plot is in Figure 30. The GRB sample number is 181. As one can see it from the figure, even this relation is not very not very tight. One would say there no much correlation between peak energies in spectra and peak photon fluxes in light curves. Alternatively, we found a more tight correlation between peak energy flux and peak energy. Peak energy flux can be calculated from peak photon flux, so this correlation is natural.
The correlation between log E p,cpl and log F pk1 is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band and in unit of 10 −6 ergs cm −2 s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.33. The Pearson coefficient is 0.48 ± 0.018 with p-value 1.7 × 10 −43 . The Spearman coefficient is 0.47 ± 0.018 with p-value 1.6 × 10 −42 . The Kendall τ coefficient is 0.33 ± 0.013 with p-value 3.4 × 10 −42 . The correlation ratio is 0.9 ± 0.0014. The cosine similarity is −0.22 ± 0.0083. The scatter plot is in Figure 31. The GRB sample number is 749. The correlation between log E p,band and log F pk2 is:

21
where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band and in unit of 10 −6 ergs cm −2 s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.33. The Pearson coefficient is 0.5 ± 0.015 with p-value 1.1 × 10 −73 . The Spearman coefficient is 0.52 ± 0.012 with p-value 6.7 × 10 −81 . The Kendall τ coefficient is 0.36 ± 0.009 with p-value 1.5 × 10 −75 . The correlation ratio is 0.89 ± 0.0024. The cosine similarity is 0.4 ± 0.015. The scatter plot is in Figure 32. The GRB sample number is 1168. One can see from these two figures, the data are still widely distributed. What's more, as the flux is the energy of the photons multiplied by the number of the photons, the flux contains the information from the spectrum. Therefore, these two correlations does not reveal more information.
5.6. Correlations about Lorentz factor Γ 0 The correlation between log Γ 0 and log E iso is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.58. The Pearson coefficient is 0.75 ± 0.016 with p-value 1.9 × 10 −10 . The Spearman coefficient is 0.68 ± 0.02 with pvalue 4.4 × 10 −8 . The Kendall τ coefficient is 0.52 ± 0.019 with p-value 9.5 × 10 −8 . The correlation ratio is 0.67 ± 0.0065. The cosine similarity is 0.79 ± 0.013. The scatter plot is in Figure 35. The positive correlation shows that the stronger bursts is likely to produce faster ejecta, which might reveals the acceleration mechanism of the central engine.
The correlation between log Γ 0 and log L pk is: where L pk is in unit of 10 52 erg s −1 and in 1-10 4 keV energy band. The adjusted R 2 is 0.66. The Pearson coefficient is 0.81 ± 0.015 with p-value 1.1 × 10 −10 . The Spearman coefficient is 0.63 ± 0.031 with p-value 7 × 10 −6 . The Kendall τ coefficient is 0.47 ± 0.027 with p-value 1 × 10 −5 . The correlation ratio is 0.77 ± 0.0061. The cosine similarity is 0.16 ± 0.026. The scatter plot is in Figure 33. The GRB sample number is 42. A previous study gave log Γ 0 = (0.28 ± 0.002) log L iso,52 + (2.39 ± 0.003) with 38 samples ). This correlation is related to eq. (45). The correlation between log Γ 0 and log t pkOpt,i is: where t pkOpt,i is in unit of s. The adjusted R 2 is 0.66. The Pearson coefficient is −0.81 ± 0.011 with p-value 1.5 × 10 −11 . The Spearman coefficient is −0.64 ± 0.019 with p-value 2.6 × 10 −6 . The Kendall τ coefficient is −0.48 ± 0.019 with p-value 3.5 × 10 −6 . The correlation ratio is 0.019 ± 0.0085. The cosine similarity is 0.9 ± 0.0016. The scatter plot is in Figure 34. The GRB sample number is 45. A previous study gave log Γ 0 = (−0.63 ± 0.04) × log t pkOpt,i + (3.69 ± 0.09) with 19 samples (Liang et al. 2010) (Notice two of them are the peak times in X-rays). The anti-correlation mainly reveals how the GRB jets decelerate by the circum burst environment. Noticing the deceleration time is also related to the profile of the density of the environment, the correlation here may not be very tight. The correlation between log Γ 0 and log t pkOpt is: where t pkOpt is in unit of s. The adjusted R 2 is 0.56. The Pearson coefficient is −0.75±0.012 with p-value 3.4×10 −9 . The Spearman coefficient is −0.59 ± 0.022 with p-value 2.4 × 10 −5 . The Kendall τ coefficient is −0.44 ± 0.021 with p-value 1.8 × 10 −5 . The correlation ratio is 0.38 ± 0.0084. The cosine similarity is 0.93 ± 0.0012. The scatter plot is in Figure 36. The GRB sample number is 45. This correlation is clearly worse than the previous one shown in equation 47, which has the redshift correction. It implies the correlation between Γ 0 and t pkOpt,i is intrinsic, and the time dilation due to redshift goes against the correlation. The correlation between log Γ 0 and log D L is: where D L is in unit of 10 28 cm. The adjusted R 2 is 0.43. The Pearson coefficient is 0.67 ± 0.01 with p-value 2.6 × 10 −8 . The Spearman coefficient is 0.5 ± 0.019 with p-value 9.6 × 10 −5 . The Kendall τ coefficient is 0.35 ± 0.014 with p-value 1.5 × 10 −4 . The correlation ratio is 0.89 ± 0.00096. The cosine similarity is 0.78 ± 0.0014. The scatter plot is in Figure 37. The GRB sample number is 55. The dependence on the distance indicates observational bias, that the low Γ 0 GRBs might not be able to be observed at longer distance. The other possibility is that the GRBs exploded earlier do have higher Γ 0 . The correlation between log Γ 0 and log t radio,pk is: log Γ 0 = (1.5 ± 0.14) × log t radio,pk + (−6.5 ± 0.81), where t radio,pk is in unit of s. The adjusted R 2 is 0.39. The Pearson coefficient is 0.64 ± 0.05 with p-value 2.4 × 10 −2 . The Spearman coefficient is 0.49 ± 0.086 with p-value 0.11. The Kendall τ coefficient is 0.37 ± 0.081 with p-value 0.1. The correlation ratio is 0.96 ± 0.0013. The cosine similarity is 0.97 ± 0.0013. The scatter plot is in Figure 38. The GRB sample number is 12. Interestingly, not like eq. (47), Γ 0 here is positively correlated with t radio,pk . As t radio,pk does not represent the deceleration time, higher Γ 0 may corresponds to higher kinetic energy, and the radio emission lasts longer. The correlation between log Γ 0 and log E p,cpl,i is: where E p,cpl,i is in unit of keV. The adjusted R 2 is 0.31. The Pearson coefficient is 0.53 ± 0.061 with p-value 9.9 × 10 −3 . The Spearman coefficient is 0.38 ± 0.087 with p-value 7.7 × 10 −2 . The Kendall τ coefficient is 0.27 ± 0.07 with p-value 6.8 × 10 −2 . The correlation ratio is 0.52 ± 0.031. The cosine similarity is 0.98 ± 0.0028. The scatter plot is in Figure 39. The GRB sample number is 23. As Γ 0 is related to E iso (as shown in eq. (45)), and E iso is related to E p (as shown in eq. (17)), it is not surprising that Γ 0 is related to E p . The correlation between log Γ 0 and log(1 + z) is: the adjusted R 2 is 0.25. The Pearson coefficient is 0.52 ± 0.024 with p-value 9.8 × 10 −5 . The Spearman coefficient is 0.42 ± 0.026 with p-value 2.3 × 10 −3 . The Kendall τ coefficient is 0.29 ± 0.018 with p-value 2.9 × 10 −3 . The correlation ratio is 0.94 ± 0.0011. The cosine similarity is 0.95 ± 0.0026. The scatter plot is in Figure 40. The GRB sample number is 51. This might be a selection effect, that further GRBs can be observed only if they are stronger and with faster jets. Otherwise, this may indicate the properties of GRBs evolve with cosmic time. This relation is intrinsically the same as shown in equation 49. The correlation between log Γ 0 and log M ass is: where Mass is in unit of M . The adjusted R 2 is 0.25. The Pearson coefficient is 0.52 ± 0.058 with p-value 3.3 × 10 −2 . The Spearman coefficient is 0.35 ± 0.1 with p-value 0.17. The Kendall τ coefficient is 0.26 ± 0.078 with p-value 0.14. The correlation ratio is 0.98 ± 0.0029. The cosine similarity is 0.98 ± 0.0017. The scatter plot is in Figure 41. The GRB sample number is 17. This weak correlation may imply that the stronger GRBs are harbored in more massive host galaxies. 5.7. Some correlations about t pkX t pkX is the peak time in the X-ray light curve. It is often taken as the deceleration time of the ejecta decelerated by the circum-burst medium. It mainly reveals the information of the initial Lorentz factor and the number density of the circum-burst medium.

23
The correlation between log F Opt11hr and log t pkX,i is: where F Opt11hr is in unit of Jy. t pkX,i is in unit of s. The adjusted R 2 is 0.36. The Pearson coefficient is 0.5 ± 0.17 with p-value 0.11. The Spearman coefficient is 0.54 ± 0.18 with p-value 8.9 × 10 −2 . The Kendall τ coefficient is 0.39 ± 0.15 with p-value 0.1. The correlation ratio is 0.98 ± 0.0049. The cosine similarity is −0.94 ± 0.014. The scatter plot is in Figure 44. The GRB sample number is 11. As the afterglow fades after the deceleration time, and the deceleration time is generally smaller than 11 hours, as shown in Figure 44, the optical afterglow might be brighter if it is more closer to the deceleration time. Therefore, F Opt11hr is roughly proportional to the t pkX,i . The correlation between log T 50 and log t pkX is: where T 50 is in unit of s. t pkX is in unit of s. The adjusted R 2 is 0.57. The Pearson coefficient is 0.74 ± 0.058 with p-value 2.4 × 10 −3 . The Spearman coefficient is 0.6 ± 0.076 with p-value 2.2 × 10 −2 . The Kendall τ coefficient is 0.47±0.074 with p-value 1.9×10 −2 . The correlation ratio is 0.52±0.023. The cosine similarity is 0.93±0.0085. The scatter plot is in Figure 45. The GRB sample number is 14. The duration is positively correlated to the deceleration time, which may indicate that the luminosity of the GRBs are kept consistent, and longer time of duration sustains the ejecta decelerated at a later time. However, as can be seen from the figure, this relation far more than a tight correlation.

Some correlations about F g
The γ−ray fluence F g is related to several other parameters of GRBs. However, as F g is just an observational properties, which is highly affected by the distance comparing with the more intrinsic quantities like the luminosity and the total energy. Therefore, the correlations about F g can not directly reveal the intrinsic properties. One would try to seek the correlations between the intrinsic properties. As the F g is one of the easiest obtained quantities, we still list the correlations in the following. One may get clues from these correlations.
The correlation between log F g and log T 50 is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. T 50 is in unit of s. The adjusted R 2 is 0.3. The Pearson coefficient is 0.53 ± 0.0044 with p-value 6.2 × 10 −211 . The Spearman coefficient is 0.53 ± 0.0032 with p-value 6.3 × 10 −212 . The Kendall τ coefficient is 0.37 ± 0.0023 with p-value 5.7 × 10 −195 . The correlation ratio is 0.15 ± 0.0021. The cosine similarity is 0.69 ± 0.0039. The scatter plot is in Figure 46. The GRB sample number is 2916. One can find two subgroups from the figure, which are defined as short GRBs and long GRBs. The correlation between log F g and log F X11hr is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. F X11hr is in unit of Jy. The adjusted R 2 is 0.29. The Pearson coefficient is 0.52 ± 0.017 with p-value 6.7 × 10 −18 . The Spearman coefficient is 0.57 ± 0.016 with p-value 2.1 × 10 −22 . The Kendall τ coefficient is 0.4 ± 0.013 with p-value 2.2 × 10 −20 . The correlation ratio is 0.98 ± 0.00041. The cosine similarity is −0.51 ± 0.012. The scatter plot is in Figure 47. The GRB sample number is 239. The correlation between log F g and log T R45,i is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. T R45,i is in unit of s. The adjusted R 2 is 0.28. The Pearson coefficient is 0.53 ± 0.0089 with p-value 1.3 × 10 −25 . The Spearman coefficient is 0.48 ± 0.0089 with p-value 5 × 10 −21 . The Kendall τ coefficient is 0.33 ± 0.0068 with p-value 5.6 × 10 −20 . The correlation ratio is 0.24 ± 0.0053. The cosine similarity is 0.64 ± 0.0064. The scatter plot is in Figure 48. The GRB sample number is 337. Interestingly, the clustering effect in this figure is not obvious comparing with Figure 46.
In Lloyd et al. (2000a), they found a correlation between F g and E p,band , and also mentioned a correlation between F g and E p,band,i . They explained this correlation is due to an intrinsic relation between the burst rest-frame peak energy and the total radiated energy. They also found the internal shock model is consistent with their interpretation of the correlation, not external shock.
The correlation between log F g and log E p,band,i is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. E p,band,i is in unit of keV. The adjusted R 2 is 0.26. The Pearson coefficient is 0.49 ± 0.027 with p-value 2.4 × 10 −12 . The Spearman coefficient is 0.51 ± 0.022 with p-value 1.8 × 10 −13 . The Kendall τ coefficient is 0.35 ± 0.016 with p-value 1.9 × 10 −12 . The correlation ratio is 0.71 ± 0.0047. The cosine similarity is 0.85 ± 0.0072. The scatter plot is in Figure 49. The GRB sample number is 179. The correlation between log F g and log T 90,i is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. T 90,i is in unit of s. The adjusted R 2 is 0.24. The Pearson coefficient is 0.48 ± 0.009 with p-value 6.9 × 10 −32 . The Spearman coefficient is 0.48 ± 0.0081 with p-value 3.2 × 10 −31 . The Kendall τ coefficient is 0.33 ± 0.006 with p-value 2.5 × 10 −29 . The correlation ratio is 0.21 ± 0.0043. The cosine similarity is 0.72 ± 0.0063. The scatter plot is in Figure 50. The GRB sample number is 526. The correlation between log F g and log T 90 is: where T 90 is in unit of s. F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. The adjusted R 2 is 0.32. The Pearson coefficient is 0.55 ± 0.004 with p-value 2.5 × 10 −276 . The Spearman coefficient is 0.55 ± 0.0029 with p-value 8.2 × 10 −281 . The Kendall τ coefficient is 0.38 ± 0.0022 with p-value 1.5 × 10 −257 . The correlation ratio is 0.38 ± 0.0018. The cosine similarity is 0.72 ± 0.0036. The scatter plot is in Figure 51. The GRB sample number is 3532. Comparing with Figure 50, the clustering effect is much more obvious in Figure 51. However, the rest frame duration should be more intrinsic. It could be a puzzle. Notice the sample number in Figure 50 is much fewer, which might cause the clustering effect not obvious. The correlation between log F g and log P pk3 is: where F g is in unit of 10 −6 ergs cm −2 and in 20-2000 keV energy band. P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.48. The Pearson coefficient is 0.65 ± 0.0073 with p-value 3.8 × 10 −304 . The Spearman coefficient is 0.64 ± 0.0049 with p-value 3 × 10 −289 . The Kendall τ coefficient is 0.46 ± 0.004 with p-value 7.7 × 10 −263 . The correlation ratio is 0.064 ± 0.0032. The cosine similarity is 0.77 ± 0.0052. The scatter plot is in Figure 52. The GRB sample number is 2531. It is quite natural that the fluence F g is proportional to peak photon numbers P pk . There are similar relations which are not shown here.

Some correlations about F X11hr
The correlation between log F X11hr and log P pk1 is: where F X11hr is in unit of Jy. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.44. The Pearson coefficient is 0.66 ± 0.026 with p-value 1.8 × 10 −4 . The Spearman coefficient is 0.68 ± 0.03 with p-value 1.1 × 10 −4 . The Kendall τ coefficient is 0.47 ± 0.027 with p-value 5.8 × 10 −4 . The correlation ratio is 0.98 ± 0.0012. The cosine similarity is −0.83 ± 0.0054. The scatter plot is in Figure 53. The GRB sample number is 27.

25
The correlation between log F X11hr and log F pk2 is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band and in unit of 10 −6 ergs cm −2 s −1 . F X11hr is in unit of Jy. The adjusted R 2 is 0.33. The Pearson coefficient is 0.55 ± 0.063 with p-value 3.8 × 10 −3 . The Spearman coefficient is 0.57 ± 0.066 with p-value 2.4 × 10 −3 . The Kendall τ coefficient is 0.39 ± 0.054 with p-value 5.1 × 10 −3 . The correlation ratio is 0.98 ± 0.0016. The cosine similarity is −0.3 ± 0.068. The scatter plot is in Figure 54. The GRB sample number is 26.

5.10
. Some correlations about F Opt11hr log F X11hr and log F Opt11hr has a remarkable linear correlation, but not very strong. The hypothesis testing pvalues of the whole formula, linear coefficient a, Pearson, Spearman and Kendall τ coefficients are all smaller than 0.05. The Pearson coefficient is 0.25 ± 0.033 with p-value 6.5 × 10 −5 . The Spearman coefficient is 0.27 ± 0.036 with p-value 1.3 × 10 −5 . The Kendall τ coefficient is 0.18 ± 0.025 with p-value 1.2 × 10 −5 . The adjusted R 2 is 0.08. Because log F X11hr has correlations with log P pk1 and log F pk2 , log F Opt11hr should have similar results.
The correlation between log F Opt11hr and log F pk2 is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.36. The Pearson coefficient is 0.51 ± 0.075 with p-value 2.5 × 10 −4 . The Spearman coefficient is 0.49 ± 0.077 with p-value 4.5 × 10 −4 . The Kendall τ coefficient is 0.35 ± 0.058 with p-value 5.3 × 10 −4 . The correlation ratio is 0.96 ± 0.0038. The cosine similarity is −0.46 ± 0.043. The scatter plot is in Figure 55. The GRB sample number is 48. The correlation between log F Opt11hr and log P pk1 is: log F Opt11hr = (0.88 ± 0.14) × log P pk1 + (−6 ± 0.17), where F Opt11hr is in unit of Jy. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.32. The Pearson coefficient is 0.49 ± 0.065 with p-value 2.8 × 10 −4 . The Spearman coefficient is 0.46 ± 0.07 with p-value 8.2 × 10 −4 . The Kendall τ coefficient is 0.33 ± 0.053 with p-value 8 × 10 −4 . The correlation ratio is 0.97 ± 0.0029. The cosine similarity is −0.85 ± 0.008. The scatter plot is in Figure  56. The GRB sample number is 50.

5.11
. Some correlations about F radio,pk The correlation between log F radio,pk and log F Opt11hr is: where F Opt11hr is in unit of Jy. F radio,pk is in unit of Jy. The adjusted R 2 is 0.32. The Pearson coefficient is 0.47 ± 0.059 with p-value 1.3 × 10 −4 . The Spearman coefficient is 0.41 ± 0.071 with p-value 1.3 × 10 −3 . The Kendall τ coefficient is 0.28 ± 0.052 with p-value 1.4 × 10 −3 . The correlation ratio is 0.56 ± 0.028. The cosine similarity is 0.99 ± 0.002. The scatter plot is in Figure 57. The GRB sample number is 60. The correlation between log F radio,pk and log SSF R is: where F radio,pk is in unit of Jy. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.34. The Pearson coefficient is 0.6 ± 0.018 with p-value 3.9 × 10 −4 . The Spearman coefficient is 0.74 ± 0.021 with p-value 2.3 × 10 −6 . The Kendall τ coefficient is 0.54 ± 0.019 with p-value 2 × 10 −5 . The correlation ratio is 0.94 ± 0.0024. The cosine similarity is 0.023 ± 0.018. The scatter plot is in Figure 58. The GRB sample number is 31.

Some correlations about α spl
The correlation between log P pk3 and (−α spl ) is: where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.29. The Pearson coefficient is 0.46 ± 0.028 with p-value 7.4 × 10 −27 . The Spearman coefficient is 0.49 ± 0.021 with p-value 2.2 × 10 −31 . The Kendall τ coefficient is 0.35 ± 0.015 with p-value 6.5 × 10 −31 . The correlation ratio is 0.87 ± 0.0027. The cosine similarity is 0.7 ± 0.018. The scatter plot is in Figure 59. The GRB sample number is 488. This might be a selection effect, that those GRBs with steep spectra (larger (−α spl )) can only be observed if they are strong, i.e., high peak photon flux. The correlation between β X11hr and (−α spl ) is: the adjusted R 2 is 0.43. The Pearson coefficient is 0.53 ± 0.14 with p-value 3.3 × 10 −2 . The Spearman coefficient is 0.52 ± 0.14 with p-value 3.7 × 10 −2 . The Kendall τ coefficient is 0.39 ± 0.11 with p-value 3.1 × 10 −2 . The correlation ratio is 0.12 ± 0.081. The cosine similarity is 0.96 ± 0.015. The scatter plot is in Figure 60. The GRB sample number is 16. It tells those GRBs with steeper spectra in the prompt emission stage also have steeper spectra in the afterglow stage, even the main emtting energy bands are different. It is likely that the emission in these two stages share the same radiation mechanism, and with the cooling of the ejecta, the radiation bands shift from γ−rays to X-rays. The correlation between log HR and (−α spl ) is: the adjusted R 2 is 0.69. The Pearson coefficient is −0.73±0.016 with p-value 1.6×10 −148 . The Spearman coefficient is −0.78 ± 0.011 with p-value 1.9 × 10 −176 . The Kendall τ coefficient is −0.6 ± 0.01 with p-value 8.4 × 10 −158 . The correlation ratio is 0.71 ± 0.0043. The cosine similarity is 0.59 ± 0.0093. The scatter plot is in Figure 61. The GRB sample number is 874. One can see a double linear relation in the figure. This may indicate that the classification is important before the correlation analysis. Because β X11hr and P pk3 both correlate with α spl , and P pk3 and P pk4 also have correlation, β X11hr and P pk4 have correlation naturally.
The correlation between log P pk4 and β X11hr is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.29. The Pearson coefficient is −0.44 ± 0.058 with p-value 1.5 × 10 −8 . The Spearman coefficient is −0.48 ± 0.037 with p-value 4.4 × 10 −10 . The Kendall τ coefficient is −0.33 ± 0.028 with p-value 2.9 × 10 −9 . The correlation ratio is 0.72 ± 0.012. The cosine similarity is 0.6 ± 0.018. The scatter plot is in Figure 62. The GRB sample number is 151.

Correlations about host galaxy offset
The correlation between log F Opt11hr and log of f set is: where F Opt11hr is in unit of Jy. Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.28. The Pearson coefficient is −0.39 ± 0.13 with p-value 4.3 × 10 −2 . The Spearman coefficient is −0.46 ± 0.12 with p-value 1.6 × 10 −2 . The Kendall τ coefficient is −0.32 ± 0.085 with p-value 1.8 × 10 −2 . The correlation ratio is 0.96 ± 0.01. The cosine similarity is −0.6 ± 0.12. The scatter plot is in Figure 63. The GRB sample number is 27. The correlation between log T 50,i and log of f set is: where host galaxy offset is in unit of kpc. T 50,i is in unit of s. The adjusted R 2 is 0.4. The Pearson coefficient is −0.56 ± 0.076 with p-value 2.6 × 10 −3 . The Spearman coefficient is −0.58 ± 0.05 with p-value 1.4 × 10 −3 . The Kendall τ coefficient is −0.42 ± 0.042 with p-value 2.2 × 10 −3 . The correlation ratio is 0.46 ± 0.05. The cosine similarity is −0.54 ± 0.043. The scatter plot is in Figure 64. The GRB sample number is 27. This is consistent with the classification of the long GRBs and the short GRBs. For short GRBs are thought occurring further from the center of the host galaxies, therefore, T 50,i and offset is anti-correlated. The correlation between spectral lag and log of f set is: where Spectral time lag is in unit of ms MeV −1 . Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.35. The Pearson coefficient is −0.53 ± 0.19 with p-value 1.4 × 10 −2 . The Spearman coefficient is −0.41 ± 0.15 with p-value 6.5 × 10 −2 . The Kendall τ coefficient is −0.3 ± 0.11 with p-value 6.1 × 10 −2 . The correlation ratio is 0.28 ± 0.051. The cosine similarity is −0.17 ± 0.18. The scatter plot is in Figure 65. The GRB sample number is 21.
The correlation between rest-frame spectral lag and log of f set is: where rest-frame spectral lag is in unit of ms MeV −1 . Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.35. The Pearson coefficient is −0.53 ± 0.19 with p-value 1.6 × 10 −2 . The Spearman coefficient is −0.35 ± 0.15 with p-value 0.12. The Kendall τ coefficient is −0.25 ± 0.11 with p-value 0.12. The correlation ratio is 0.29 ± 0.05. The cosine similarity is −0.17 ± 0.19. The scatter plot is in Figure 66. The GRB sample number is 20. This may also comes from the difference of long GRBs and short GRBs. Short GRBs are thought having smaller spectral lags.
Noticing the large scattering, one may not get much information from it.
5.14. Some correlations about t radio,pk The correlation between rest-frame spectral lag and log t radio,pk,i is: where rest-frame spectral lag is in unit of ms MeV −1 . t radio,pk,i is in unit of s. The adjusted R 2 is 0.34. The Pearson coefficient is 0.53 ± 0.072 with p-value 1.3 × 10 −3 . The Spearman coefficient is 0.38 ± 0.076 with p-value 2.7 × 10 −2 . The Kendall τ coefficient is 0.27 ± 0.056 with p-value 2.6 × 10 −2 . The correlation ratio is 0.33 ± 0.061. The cosine similarity is 0.48 ± 0.075. The scatter plot is in Figure 67. The GRB sample number is 33. This is an interesting correlations. The spectral lag is a quantity of the prompt emission, while the physical origin is still not clear. The peak time of the radio emission t radio,pk is a quantity of the late afterglow. The spectral lag might be related to the radiation mechanism, while t radio,pk is more likely related to the total energy and the environment. They are not likely to be related to each other. The reason of this correlation is not clear. The correlation between log P pk1 and log t radio,pk is: log P pk1 = (−0.8 ± 0.066) × log t radio,pk + (6.1 ± 0.38), where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . t radio,pk is in unit of s. The adjusted R 2 is 0.31. The Pearson coefficient is −0.59 ± 0.032 with p-value 1.3 × 10 −2 . The Spearman The correlation ratio is 0.98 ± 0.0011. The cosine similarity is 0.92 ± 0.0077. The scatter plot is in Figure 68. The GRB sample number is 17. The correlation between log F pk2 and log t radio,pk is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . t radio,pk is in unit of s. The adjusted R 2 is 0.27. The Pearson coefficient is −0.54 ± 0.049 with p-value 2.5 × 10 −2 . The Spearman coefficient is −0.41 ± 0.049 with p-value 0.11. The Kendall τ coefficient is −0.3 ± 0.043 with p-value 9.9 × 10 −2 . The correlation ratio is 0.98 ± 0.002. The cosine similarity is 0.77 ± 0.044. The scatter plot is in Figure 69. The GRB sample number is 17.

Correlations about N H
The correlation between log N H and log Age is: where N H is in unit of 10 21 cm −2 . Age is in unit of Myr. The adjusted R 2 is 0.25. The Pearson coefficient is −0.49 ± 0.077 with p-value 1.2 × 10 −2 . The Spearman coefficient is −0.45 ± 0.083 with p-value 2.2 × 10 −2 . The Kendall τ coefficient is −0.32 ± 0.067 with p-value 2.1 × 10 −2 . The correlation ratio is 0.78 ± 0.034. The cosine similarity is 0.53 ± 0.055. The scatter plot is in Figure 70. The GRB sample number is 26. This may reveals the property of the evolution of the host galaxies, that more material has been formed into stars for the older galaxies. The correlation between log N H and log SF R is: where N H is in unit of 10 21 cm −2 . SFR is in unit of M yr −1 . The adjusted R 2 is 0.31. The Pearson coefficient is 0.55 ± 0.056 with p-value 5.6 × 10 −4 . The Spearman coefficient is 0.52 ± 0.056 with p-value 1.3 × 10 −3 . The Kendall τ coefficient is 0.37 ± 0.046 with p-value 1.6 × 10 −3 . The correlation ratio is 0.03 ± 0.022. The cosine similarity is 0.71 ± 0.034. The scatter plot is in Figure 71. The GRB sample number is 36. It is clear that in a star forming galaxy, the material must be rich. Consequently, the column density of hydrogen is high. The correlation between log N H and log F pk4 is: where F pk4 is peak energy flux of 1024 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.35. The Pearson coefficient is −0.58 ± 0.055 with p-value 3.1 × 10 −3 . The Spearman coefficient is −0.58 ± 0.078 with p-value 3 × 10 −3 . The Kendall τ coefficient is −0.43 ± 0.061 with p-value 3.4 × 10 −3 . The correlation ratio is 0.45 ± 0.027. The cosine similarity is −0.16 ± 0.05. The scatter plot is in Figure 72. The GRB sample number is 24. The correlation between log N H and log P pk3 is: where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.36. The Pearson coefficient is −0.59 ± 0.049 with p-value 2.7 × 10 −3 . The Spearman coefficient is −0.61 ± 0.058 with p-value 1.5 × 10 −3 . The Kendall τ coefficient is −0.45 ± 0.05 with p-value 2.1 × 10 −3 . The correlation ratio is 0.1 ± 0.031. The cosine similarity is 0.49 ± 0.026. The scatter plot is in Figure  73. The GRB sample number is 24.

Correlations about Age of the host galaxies
The correlation between log Age and log t burst,i is: where Age is in unit of Myr. t burst,i is in unit of s. The adjusted R 2 is 0.3. The Pearson coefficient is 0.54 ± 0.093 with p-value 2 × 10 −2 . The Spearman coefficient is 0.69 ± 0.094 with p-value 1.6 × 10 −3 . The Kendall τ coefficient is 0.51 ± 0.08 with p-value 3.5 × 10 −3 . The correlation ratio is 0.044 ± 0.032. The cosine similarity is 0.96 ± 0.0073. The scatter plot is in Figure 74. The GRB sample number is 18. t burst,i indicates the active time scale of the GRB central engine. It is unclear why the older host galaxies contain GRBs with longer activity. The correlation between log Age and log SF R is: where SFR is in unit of M yr −1 . Age is in unit of Myr. The adjusted R 2 is 0.28. The Pearson coefficient is −0.5 ± 0.067 with p-value 6.4 × 10 −3 . The Spearman coefficient is −0.49 ± 0.079 with p-value 7.6 × 10 −3 . The Kendall τ coefficient is −0.35 ± 0.062 with p-value 9.1 × 10 −3 . The correlation ratio is 0.8 ± 0.014. The cosine similarity is 0.27 ± 0.035. The scatter plot is in Figure 75. The GRB sample number is 28. It is natural that star forming is more active in a younger galaxy. The correlation between the age and the SFR is an evidence. The correlation between log Age and log D L is: where D L is in unit of 10 28 cm. Age is in unit of Myr. The adjusted R 2 is 0.26. The Pearson coefficient is −0.3 ± 0.081 with p-value 6.9 × 10 −3 . The Spearman coefficient is −0.2 ± 0.091 with p-value 7.5 × 10 −2 . The Kendall τ coefficient is −0.14 ± 0.064 with p-value 7.3 × 10 −2 . The correlation ratio is 0.99 ± 0.002. The cosine similarity is −0.82 ± 0.0058. The scatter plot is in Figure 76. The GRB sample number is 82. The correlation between log Age and log(1 + z) is: where Age is in unit of Myr. The adjusted R 2 is 0.22. The Pearson coefficient is −0.39±0.059 with p-value 3×10 −4 . The Spearman coefficient is −0.43 ± 0.049 with p-value 5.5 × 10 −5 . The Kendall τ coefficient is −0.29 ± 0.035 with p-value 9.9 × 10 −5 . The correlation ratio is 0.91 ± 0.011. The cosine similarity is 0.76 ± 0.03. The scatter plot is in Figure 77. The GRB sample number is 82. It is similar as shown in eq. (86). It is quite reasonable that the further galaxy is younger. However, the relation between age of the host galaxies and the redshift is not exactly because of the cosmological evolution. Considering the large scattering, it is hard to tell the special information of the GRB host galaxies from the general evolution of the universe. Age also correlates with E p,cpl and E p,cpl,i . We have shown the two correlations in section 5.6.

Some correlations about L radio,pk
We calculated radio luminosity L radio,pk in rest-frame 8.46 GHz. The formula is L radio,pk = 4πD 2 L × F radio,pk × (1 + z) × 8.46 GHz. The unit of L radio,pk is 10 40 erg s −1 . We also found some interesting results.
The correlation between log L radio,pk and log E iso is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.41. The Pearson coefficient is 0.64 ± 0.0094 with p-value 2.2 × 10 −8 . The Spearman coefficient is 0.52 ± 0.015 with p-value 1.5 × 10 −5 . The Kendall τ coefficient is 0.37 ± 0.013 with p-value 2.6 × 10 −5 . The correlation ratio is 0.37 ± 0.0057. The cosine similarity is 0.81 ± 0.0067. The scatter plot is in Figure 78. The GRB sample number is 61. The peak time of the radio emission are often very late. The peak radio luminosity mainly represents the total kinetic energy and environment, while the isotropic equivalent γ−ray energy E iso is also positively correlated to the total kinetic energy, though people like to assume they are proportional by an efficiency factor. Therefore, it is expectable about the correlation between L radio,pk and E iso . As the data are largely scattered, the actually power law index between them is not clear. With cumulated data and classification, the indices might change.
The correlation between log L radio,pk and log L pk is: log L radio,pk = (0.52 ± 0.025) × log L pk + (1.7 ± 0.018), where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.45. The Pearson coefficient is 0.67 ± 0.023 with p-value 9.1 × 10 −8 . The Spearman coefficient is 0.45 ± 0.029 with p-value 8.2 × 10 −4 . The Kendall τ coefficient is 0.32 ± 0.021 with p-value 8.9 × 10 −4 . The correlation ratio is 0.57 ± 0.0055. The cosine similarity is 0.4 ± 0.027. The scatter plot is in Figure  79. The GRB sample number is 51. This relation is similar to eq. (88). The correlation between log L radio,pk and log SF R is: log L radio,pk = (0.63 ± 0.092) × log SF R + (0.43 ± 0.073), where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . SFR is in unit of M yr −1 . The adjusted R 2 is 0.25. The Pearson coefficient is 0.53 ± 0.057 with p-value 4 × 10 −2 . The Spearman coefficient is 0.37 ± 0.06 with p-value 0.17. The Kendall τ coefficient is 0.25 ± 0.052 with p-value 0.18. The correlation ratio is 0.13 ± 0.03. The cosine similarity is 0.64 ± 0.054. The scatter plot is in Figure 80. The GRB sample number is 15. This relation may reveal the dependance of the radio afterglow and the environment of the GRBs. Higher star formation rate region implies higher number density of the environment. The correlation between log L radio,pk and log Age is: log L radio,pk = (−0.61 ± 0.16) × log Age + (2.9 ± 0.48), where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . Age is in unit of Myr. The adjusted R 2 is 0.21. The Pearson coefficient is −0.43 ± 0.085 with p-value 3.4 × 10 −2 . The Spearman coefficient is −0.44 ± 0.099 with p-value 3.1 × 10 −2 . The Kendall τ coefficient is −0.31 ± 0.077 with p-value 3.3 × 10 −2 . The correlation ratio is 0.66 ± 0.038. The cosine similarity is 0.64 ± 0.024. The scatter plot is in Figure 81. The GRB sample number is 24. As shown in eq. (90), higher radio luminosity corresponds to more active star forming. And in eq. (85), higher star formation rate corresponds to the younger host galaxy. These two relations derives to eq. (91). The correlation between log L radio,pk and log E p,band,i is: log L radio,pk = (0.86 ± 0.054) × log E p,band,i + (−0.48 ± 0.14), where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,band,i is in unit of keV. The adjusted R 2 is 0.32. The Pearson coefficient is 0.57 ± 0.027 with p-value 2.6 × 10 −4 . The Spearman coefficient is 0.5 ± 0.037 with p-value 1.8 × 10 −3 . The Kendall τ coefficient is 0.35 ± 0.031 with p-value 2.5 × 10 −3 . The correlation ratio is 0.51 ± 0.011. The cosine similarity is 0.93 ± 0.0032. The scatter plot is in Figure 82. The GRB sample number is 36. This relation might be from the Amati relation. As higher peak energy corresponds to higher total energy of the GRBs, it also derives to a brighter radio luminosity. The correlation between log L radio,pk and log E p,cpl,i is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,cpl,i is in unit of keV. The adjusted R 2 is 0.42. The Pearson coefficient is 0.65 ± 0.057 with p-value 2.1 × 10 −3 . The Spearman coefficient is 0.57 ± 0.077 with p-value 8.7 × 10 −3 . The Kendall τ coefficient is 0.42 ± 0.064 with p-value 9.4 × 10 −3 . The correlation ratio is 0.31 ± 0.018. The cosine similarity is 0.88 ± 0.0075. The scatter plot is in Figure 83. The GRB sample number is 20. It is similar to eq. (92). The correlation between log L radio,pk and rest-frame spectral lag is: log L radio,pk = (−0.00023 ± 0.000048) × rest − f rame spectral lag + (2 ± 0.063), where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . Rest-frame spectral lag is in unit of ms MeV −1 . The adjusted R 2 is 0.29. The Pearson coefficient is −0.49 ± 0.089 with p-value 4.2 × 10 −3 . The Spearman coefficient is −0.4 ± 0.081 with p-value 2.2 × 10 −2 . The Kendall τ coefficient is −0.28 ± 0.06 with p-value 2.3 × 10 −2 . The correlation ratio is 0.34 ± 0.06. The cosine similarity is 0.2 ± 0.078. The scatter plot is in Figure 84. The GRB sample number is 32. It is also quite puzzle why the radio luminosity if related to the spectral lag.

REMARKABLE RESULTS OF THREE PARAMETERS
In this section, we analyzed some good correlations between three parameters. Firstly the correlation should also have at least 10 GRBs. Adjusted R 2 should also be bigger than 0.2. The whole linear model, a 1 and a 2 have hypothesis testing p-values smaller than 0.05. For peak energy flux F pk and peak photon flux P pk , both of them have four different time bins. If F pk in all the four time bins correlates with another one same parameter, we just showed the best time bin result. It is the same for P pk . All the results are in machine readable tables. In the following, we will show all the remarkable results in order of increasing sample numbers.

31
The log SSF R-log M ass-log t pkX,i formula is: where log SSF R is in unit of Gyr −1 . Mass is in unit of M . t pkX,i is in unit of s. The adjusted R 2 is 0.6759. The GRB sample number is 10. The log t pkX,i -log SSF R-log T 90,i formula is: where t pkX,i is in unit of s. log SSF R is in unit of Gyr −1 . T 90,i is in unit of s. The adjusted R 2 is 0.7597. The GRB sample number is 10. The log Age-log N H -log t burst,i formula is: where Age is in unit of Myr. N H is in unit of 10 21 cm −2 . t burst,i is in unit of s. The adjusted R 2 is 0.681. The GRB sample number is 10. The (−β band )-log of f set-log Age formula is: where host galaxy offset is in unit of kpc. Age is in unit of Myr. The adjusted R 2 is 0.5508. The GRB sample number is 10.
The log E iso -log t pkOpt -log Age formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. t pkOpt is in unit of s. Age is in unit of Myr. The adjusted R 2 is 0.9079. The GRB sample number is 10. The log Γ 0 -log L radio,pk -log T R45 formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . T R45 is in unit of s. The adjusted R 2 is 0.6438. The GRB sample number is 10. The β X11hr -(−α spl )-log M ass formula is: where Mass is in unit of M . The adjusted R 2 is 0.5919. The GRB sample number is 11. The log E p,cpl -log HR-log t pkX formula is: where E p,cpl is in unit of keV. t pkX is in unit of s. The adjusted R 2 is 0.9356. The GRB sample number is 11.
The log E iso -log t burst,i -log t radio,pk,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. t burst,i is in unit of s. t radio,pk,i is in unit of s. The adjusted R 2 is 0.6305. The GRB sample number is 11. The log F radio,pk -log t burst,i -log t radio,pk,i formula is: log F radio,pk = (0.24 ± 0.026) × log t burst,i + (0.66 ± 0.055) × log t radio,pk,i + (−7.7 ± 0.31), where F radio,pk is in unit of Jy. t burst,i is in unit of s. t radio,pk,i is in unit of s. The adjusted R 2 is 0.4834. The GRB sample number is 11.
The β X11hr -log P pk2 -log SSF R formula is: where P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.6811. The GRB sample number is 11. The log Γ 0 -log t radio,pk -log T 50,i formula is: where t radio,pk is in unit of s. T 50,i is in unit of s. The adjusted R 2 is 0.6372. The GRB sample number is 11. The log M ass-log T 90 -log t pkX formula is: where Mass is in unit of M . T 90 is in unit of s. t pkX is in unit of s. The adjusted R 2 is 0.7206. The GRB sample number is 11. The log L pk -log L radio,pk -log t burst formula is: where L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t burst is in unit of s. The adjusted R 2 is 0.8403. The GRB sample number is 11. The rest-frame spectral lag-log L radio,pk -log E p,cpl formula is: where rest-frame spectral lag is in unit of ms MeV −1 . L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.7246. The GRB sample number is 11. The log Γ 0 -log L radio,pk -log t pkOpt formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t pkOpt is in unit of s. The adjusted R 2 is 0.9562. The GRB sample number is 11.
The log E iso -log Γ 0 -log F radio,pk formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F radio,pk is in unit of Jy. The adjusted R 2 is 0.8299. The GRB sample number is 12.
The log F g -log Γ 0 -log t radio,pk,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. t radio,pk,i is in unit of s. The adjusted R 2 is 0.7167. The GRB sample number is 12. The Mag-log F Opt11hr -metallicity formula is: where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.5847. The GRB sample number is 13. The log L pk -log of f set-log E p,cpl,i formula is:

33
where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Host galaxy offset is in unit of kpc. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.4724. The GRB sample number is 13. The log t pkX -log T 50 -log HR formula is: where t pkX is in unit of s. T 50 is in unit of s. The adjusted R 2 is 0.7483. The GRB sample number is 13. The log F pk3 -log T 90 -log F X11hr formula is: where F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . T 90 is in unit of s. F X11hr is in unit of Jy. The adjusted R 2 is 0.7421. The GRB sample number is 13. The log P pk2 -log T 90 -log F X11hr formula is: where P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . T 90 is in unit of s. F X11hr is in unit of Jy. The adjusted R 2 is 0.65. The GRB sample number is 13. The (−α spl )-variability 1 -log F X11hr formula is: where F X11hr is in unit of Jy. The adjusted R 2 is 0.4567. The GRB sample number is 13. The log F pk2 -(−β band )-log t radio,pk formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . t radio,pk is in unit of s. The adjusted R 2 is 0.499. The GRB sample number is 14. The (−β band )-log F pk2 -log F radio,pk formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . F radio,pk is in unit of Jy. The adjusted R 2 is 0.6076. The GRB sample number is 14.
The log HR-log F pk3 -log Age formula is: where F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . Age is in unit of Myr. The adjusted R 2 is 0.5974. The GRB sample number is 14.
The log t pkOpt -log L pk -log SSF R formula is: where t pkOpt is in unit of s. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.8097. The GRB sample number is 14.
where t pkOpt,i is in unit of s. Mag is in unit of magnitude. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4411. The GRB sample number is 14. The (−β band )-log P pk1 -log F radio,pk formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F radio,pk is in unit of Jy. The adjusted R 2 is 0.5843. The GRB sample number is 14.
The log HR-log P pk2 -log Age formula is: where P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . Age is in unit of Myr. The adjusted R 2 is 0.3174. The GRB sample number is 14.
The log L pk -log L radio,pk -log t pkOpt formula is: where L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t pkOpt is in unit of s. The adjusted R 2 is 0.8128. The GRB sample number is 14.
The log Age-log θ j -log N H formula is: where Age is in unit of Myr. θ j is in unit of rad. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.3373. The GRB sample number is 15.
where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.5037. The GRB sample number is 16.
where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,cpl is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.6483. The GRB sample number is 16. The log Γ 0 -log F g -log M ass formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.6784. The GRB sample number is 16. The (−α band )-variability 3 -log P pk4 formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.4653. The GRB sample number is 16.
The log E p,cpl,i -log L radio,pk -A V formula is: where E p,cpl,i is in unit of keV. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . The adjusted R 2 is 0.5624. The GRB sample number is 16. The log F g -(−α spl )-log F X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F X11hr is in unit of Jy. The adjusted R 2 is 0.4784. The GRB sample number is 17.

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The log Γ 0 -log F pk1 -log M ass formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 .
Mass is in unit of M . The adjusted R 2 is 0.4798. The GRB sample number is 17. The log(1 + z)-log F pk2 -log F radio,pk formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . F radio,pk is in unit of Jy. The adjusted R 2 is 0.6147. The GRB sample number is 17. The log t radio,pk -log F radio,pk -log t pkOpt,i formula is: where t radio,pk is in unit of s. F radio,pk is in unit of Jy. t pkOpt,i is in unit of s. The adjusted R 2 is 0.3468. The GRB sample number is 17. The log D L -log Γ 0 -log M ass formula is: where D L is in unit of 10 28 cm. Mass is in unit of M . The adjusted R 2 is 0.7751. The GRB sample number is 17. The spectral lag-log t burst,i -log t pkOpt,i formula is: where spectral time lag is in unit of ms where θ j is in unit of rad. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.6444. The GRB sample number is 17. The log L radio,pk -log T 90 -log t pkOpt,i formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . T 90 is in unit of s. t pkOpt,i is in unit of s. The adjusted R 2 is 0.5211. The GRB sample number is 17.
The log E iso -log L radio,pk -log t pkOpt,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t pkOpt,i is in unit of s. The adjusted R 2 is 0.6016. The GRB sample number is 17. The log T R45 -log F X11hr -log t radio,pk,i formula is: where T R45 is in unit of s. F X11hr is in unit of Jy. t radio,pk,i is in unit of s. The adjusted R 2 is 0.3888. The GRB sample number is 18. The (−β band )-log P pk1 -log SSF R formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.4079. The GRB sample number is 18.
The log t radio,pk,i -log T 50 -log SSF R formula is: where t radio,pk,i is in unit of s. T 50 is in unit of s. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.6841. The GRB sample number is 18. The log θ j -variability 3 -log N H formula is: where θ j is in unit of rad. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4522. The GRB sample number is 18. The metallicity-(−α cpl )-β X11hr formula is: where metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.3058. The GRB sample number is 19.
The log E p,cpl,i -log HR-log t radio,pk,i formula is: where E p,cpl,i is in unit of keV. t radio,pk,i is in unit of s. The adjusted R 2 is 0.6582. The GRB sample number is 19. The log L pk -log t pkOpt -Mag formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. t pkOpt is in unit of s. Mag is in unit of magnitude. The adjusted R 2 is 0.9072. The GRB sample number is 19. The Mag-log(1 + z)-metallicity formula is: where Mag is in unit of magnitude. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.7636. The GRB sample number is 19. The log L radio,pk -log F Opt11hr -log E p,cpl,i formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . F Opt11hr is in unit of Jy. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.6282. The GRB sample number is 19.
where F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of where F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3106. The GRB sample number is 20. The log Γ 0 -log F Opt11hr -log E p,cpl,i formula is: where F Opt11hr is in unit of Jy. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.5677. The GRB sample number is 20.

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The log D L -log F radio,pk -log E p,cpl,i formula is: where D L is in unit of 10 28 cm. F radio,pk is in unit of Jy. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.5231. The GRB sample number is 20. The log SF R-log HR-(−α spl ) formula is: where SFR is in unit of M yr −1 . The adjusted R 2 is 0.2678. The GRB sample number is 20. The log SF R-Mag-log E p,cpl,i formula is: where SFR is in unit of M yr −1 . Mag is in unit of magnitude. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.8205. The GRB sample number is 20. The log SF R-log(1 + z)-(−α spl ) formula is: where SFR is in unit of M yr −1 . The adjusted R 2 is 0.4955. The GRB sample number is 20.
The log E iso -log L radio,pk -log E p,cpl formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.7599. The GRB sample number is 20. The log T 90,i -log L radio,pk -(−α cpl ) formula is: where T 90,i is in unit of s. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . The adjusted R 2 is 0.3241. The GRB sample number is 20. The log t pkOpt -log E p,band -log t burst formula is: where t pkOpt is in unit of s. E p,band is in unit of keV. t burst is in unit of s. The adjusted R 2 is 0.2739. The GRB sample number is 21. The log P pk1 -log F g -metallicity formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.5787. The GRB sample number is 21. The spectral lag-log(1 + z)-log F pk3 formula is: where spectral time lag is in unit of ms MeV −1 . F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.4887. The GRB sample number is 21. The spectral lag-log(1 + z)-log P pk2 formula is: where spectral time lag is in unit of ms MeV −1 . P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.3876. The GRB sample number is 21. The log F radio,pk -(−α band )-log M ass formula is: where F radio,pk is in unit of Jy. Mass is in unit of M . The adjusted R 2 is 0.434. The GRB sample number is 22. The log F pk2 -(−α cpl )-log F Opt11hr formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3327. The GRB sample number is 22. The log SF R-metallicity-log Age formula is: where SFR is in unit of M yr −1 . Metallicity is the value of 12 where Mag is in unit of magnitude. N H is in unit of 10 21 cm −2 . E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2554. The GRB sample number is 22.
The log E iso -log of f set-log E p,cpl,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Host galaxy offset is in unit of kpc. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.5009. The GRB sample number is 22. The Mag-log SF R-log t burst,i formula is: where Mag is in unit of magnitude. SFR is in unit of M yr −1 . t burst,i is in unit of s. The adjusted R 2 is 0.8763. The GRB sample number is 22. The log F radio,pk -log(1 + z)-variability 1 formula is: where F radio,pk is in unit of Jy. The adjusted R 2 is 0.286. The GRB sample number is 22. The log T R45,i -log L radio,pk -log SSF R formula is: where T R45,i is in unit of s. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.2284. The GRB sample number is 23. The log F X11hr -log F pk2 -β X11hr formula is: where F X11hr is in unit of Jy. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.6052. The GRB sample number is 23.
The log E iso -log Γ 0 -log E p,cpl,i formula is:

39
where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.9021. The GRB sample number is 23.
The log E p,band,i -log HR-log SF R formula is: where E p,band,i is in unit of keV. SFR is in unit of M yr −1 . The adjusted R 2 is 0.6949. The GRB sample number is 23. The Mag-log HR-log E p,band,i formula is: where Mag is in unit of magnitude. E p,band,i is in unit of keV. The adjusted R 2 is 0.3913. The GRB sample number is 23.
The log of f set-log T 50 -log P pk4 formula is: where host galaxy offset is in unit of kpc. T 50 is in unit of s. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.4552. The GRB sample number is 23. The log T 90 -variability 2 -log E p,cpl,i formula is: where T 90 is in unit of s. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.3912. The GRB sample number is 23.
The log E iso -log L radio,pk -variability 3 formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . The adjusted R 2 is 0.3608. The GRB sample number is 24. The Mag-(−β band )-log F Opt11hr formula is: where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3541. The GRB sample number is 24. The log P pk1 -log F g -log N H formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.6225. The GRB sample number is 24. The log t pkOpt -log N H -log t burst,i formula is: where t pkOpt is in unit of s. N H is in unit of 10 21 cm −2 . t burst,i is in unit of s. The adjusted R 2 is 0.3136. The GRB sample number is 24. The log F Opt11hr -log SF R-log t burst,i formula is: where F Opt11hr is in unit of Jy. SFR is in unit of M yr −1 . t burst,i is in unit of s. The adjusted R 2 is 0.4728. The GRB sample number is 24.
The log F pk3 -log(1 + z)-log N H formula is: where F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4391. The GRB sample number is 24.
The log SF R-log F Opt11hr -log N H formula is: where SFR is in unit of M yr −1 . F Opt11hr is in unit of Jy. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.5387. The GRB sample number is 25.
where F pk4 is peak energy flux of 1024 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.2497. The GRB sample number is 25. The Mag-log L pk -(−β band ) formula is: where Mag is in unit of magnitude. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.227. The GRB sample number is 25. The log F Opt11hr -log M ass-log t pkOpt,i formula is: where F Opt11hr is in unit of Jy. Mass is in unit of M . t pkOpt,i is in unit of s. The adjusted R 2 is 0.3158. The GRB sample number is 25. The A V -log P pk3 -(−α cpl ) formula is: where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2262. The GRB sample number is 25. The log N H -log T 90 -log F pk2 formula is: where N H is in unit of 10 21 cm −2 . T 90 is in unit of s. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.4444. The GRB sample number is 25. The rest-frame spectral lag-variability 3 -log θ j formula is: where rest-frame spectral lag is in unit of ms MeV −1 . θ j is in unit of rad. The adjusted R 2 is 0.4137. The GRB sample number is 25. The log SF R-A V -log Age formula is: where SFR is in unit of M yr −1 . Age is in unit of Myr. The adjusted R 2 is 0.4286. The GRB sample number is 26.
The log N H -log Age-log T 90,i formula is:

41
where N H is in unit of 10 21 cm −2 . Age is in unit of Myr. T 90,i is in unit of s. The adjusted R 2 is 0.4491. The GRB sample number is 26. The log Age-log E iso -log N H formula is: where Age is in unit of Myr. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.3666. The GRB sample number is 26. The log Age-log(1 + z)-log N H formula is: where Age is in unit of Myr. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.5434. The GRB sample number is 26. The log T 90,i -β X11hr -log of f set formula is: where T 90,i is in unit of s. Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.4588. The GRB sample number is 27. The log T 50 -log F g -log of f set formula is: where T 50 is in unit of s. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.5799. The GRB sample number is 27.
where F Opt11hr is in unit of Jy. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.406. The GRB sample number is 27. The log F g -log F X11hr -log Age formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F X11hr is in unit of Jy. Age is in unit of Myr. The adjusted R 2 is 0.6765. The GRB sample number is 27. The Mag-log L pk -log SF R formula is: where Mag is in unit of magnitude. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. SFR is in unit of M yr −1 . The adjusted R 2 is 0.837. The GRB sample number is 27. The log F Opt11hr -metallicity-log Age formula is: where F Opt11hr is in unit of Jy. Metallicity is the value of 12 + log O/H. Age is in unit of Myr. The adjusted R 2 is 0.471. The GRB sample number is 27. The log M ass-log of f set-log Age formula is: log M ass = (0.42 ± 0.15) × log of f set + (0.69 ± 0.15) × log Age + (7.9 ± 0.37), where Mass is in unit of M . Host galaxy offset is in unit of kpc. Age is in unit of Myr. The adjusted R 2 is 0.6363. The GRB sample number is 27.
The log SF R-log Age-log M ass formula is: where SFR is in unit of M yr −1 . Age is in unit of Myr. Mass is in unit of M . The adjusted R 2 is 0.6711. The GRB sample number is 28. The log T 90 -β X11hr -log of f set formula is: where T 90 is in unit of s. Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.5046. The GRB sample number is 28. The log M ass-log E p,band -log SF R formula is: where Mass is in unit of M . E p,band is in unit of keV. SFR is in unit of M yr −1 . The adjusted R 2 is 0.5525. The GRB sample number is 28.
where F Opt11hr is in unit of Jy. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.4429. The GRB sample number is 28. The log F g -log Γ 0 -log F X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F X11hr is in unit of Jy. The adjusted R 2 is 0.2285. The GRB sample number is 28. The log M ass-log P pk1 -log Age formula is: where Mass is in unit of M . P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . Age is in unit of Myr. The adjusted R 2 is 0.3665. The GRB sample number is 28. The log HR-(−α band )-log F radio,pk formula is: where F radio,pk is in unit of Jy. The adjusted R 2 is 0.4247. The GRB sample number is 29. The log M ass-(−β band )-metallicity formula is: where Mass is in unit of M . Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.2659. The GRB sample number is 29. The spectral lag-log F pk2 -log T R45,i formula is: where spectral time lag is in unit of ms MeV −1 . F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . T R45,i is in unit of s. The adjusted R 2 is 0.2428. The GRB sample number is 29. The log F Opt11hr -log F pk3 -log T 50,i formula is: where F Opt11hr is in unit of Jy. F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . T 50,i is in unit of s. The adjusted R 2 is 0.3968. The GRB sample number is 29. The log M ass-metallicity-log E p,band,i formula is: where Mass is in unit of M . Metallicity is the value of 12 + log O/H. E p,band,i is in unit of keV. The adjusted R 2 is 0.2796. The GRB sample number is 29. The log F Opt11hr -log P pk2 -log T 50,i formula is: where F Opt11hr is in unit of Jy. P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . T 50,i is in unit of s. The adjusted R 2 is 0.2895. The GRB sample number is 29. The Mag-log D L -(−β band ) formula is: where Mag is in unit of magnitude. D L is in unit of 10 28 cm. The adjusted R 2 is 0.2703. The GRB sample number is 30. where where Mag is in unit of magnitude. T 90 is in unit of s. The adjusted R 2 is 0.3291. The GRB sample number is 30. The log Γ 0 -log θ j -log T 50,i formula is: where θ j is in unit of rad. T 50,i is in unit of s. The adjusted R 2 is 0.2545. The GRB sample number is 30. The log P pk3 -log(1 + z)-log t pkOpt formula is: where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . t pkOpt is in unit of s. The adjusted R 2 is 0.5827. The GRB sample number is 30. The Mag-log(1 + z)-(−β band ) formula is: where Mag is in unit of magnitude. The adjusted R 2 is 0.3315. The GRB sample number is 30. The log L pk -log L radio,pk -log E p,band formula is: where L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.5213. The GRB sample number is 30. The log M ass-log L radio,pk -log SSF R formula is: log M ass = (0.36 ± 0.07) × log L radio,pk + (−0.52 ± 0.18) × log SSF R + (9 ± 0.086), where Mass is in unit of M . L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.3256. The GRB sample number is 30. The log(1 + z)-log Γ 0 -log N H formula is: where N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.481. The GRB sample number is 31. The Mag-log SSF R-log M ass formula is: where Mag is in unit of magnitude. log SSF R is in unit of Gyr −1 . Mass is in unit of M . The adjusted R 2 is 0.8329. The GRB sample number is 31.
where t pkOpt,i is in unit of s. F Opt11hr is in unit of Jy. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4855. The GRB sample number is 32. The log SF R-log N H -log M ass formula is: where SFR is in unit of M yr −1 . N H is in unit of 10 21 cm −2 . Mass is in unit of M . The adjusted R 2 is 0.6262. The GRB sample number is 32. The log L pk -log P pk4 -log F radio,pk formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F radio,pk is in unit of Jy. The adjusted R 2 is 0.2764. The GRB sample number is 32. The (−α band )-variability 3 -A V formula is: the adjusted R 2 is 0.2283. The GRB sample number is 32. The rest-frame spectral lag-log L radio,pk -log t radio,pk formula is: rest − f rame spectral lag = (−1029 ± 290) × log L radio,pk + (1663 ± 355) × log t radio,pk + (−7091 ± 1663), (229) where rest-frame spectral lag is in unit of ms MeV −1 . L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t radio,pk is in unit of s. The adjusted R 2 is 0.4842. The GRB sample number is 32. The log L radio,pk -A V -log E p,band,i formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,band,i is in unit of keV. The adjusted R 2 is 0.4928. The GRB sample number is 32. The log HR-log E p,cpl -log Age formula is: where E p,cpl is in unit of keV. Age is in unit of Myr. The adjusted R 2 is 0.9192. The GRB sample number is 33. The log F Opt11hr -log HR-log P pk2 formula is: where F Opt11hr is in unit of Jy. P pk2 is peak photon flux of 256 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2795. The GRB sample number is 33. The Mag-log SF R-log M ass formula is: where Mag is in unit of magnitude. SFR is in unit of M yr −1 . Mass is in unit of M . The adjusted R 2 is 0.8705. The GRB sample number is 33. The log L radio,pk -A V -log M ass formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . Mass is in unit of M . The adjusted R 2 is 0.2234. The GRB sample number is 34. The log L pk -β X11hr -log t pkOpt,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. t pkOpt,i is in unit of s. The adjusted R 2 is 0.2613. The GRB sample number is 34.
where Age is in unit of Myr. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.4063. The GRB sample number is 34. The log P pk1 -log L pk -log F Opt11hr formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.4753. The GRB sample number is 34. The log F g -variability 3 -log T 50,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T 50,i is in unit of s. The adjusted R 2 is 0.388. The GRB sample number is 34. The log Age-log D L -log E p,cpl formula is: where Age is in unit of Myr. D L is in unit of 10 28 cm. E p,cpl is in unit of keV. The adjusted R 2 is 0.4641. The GRB sample number is 35. The rest-frame spectral lag-log D L -log SF R formula is: where rest-frame spectral lag is in unit of ms MeV −1 . D L is in unit of 10 28 cm. SFR is in unit of M yr −1 . The adjusted R 2 is 0.384. The GRB sample number is 35. The log SF R-log E iso -log N H formula is: where SFR is in unit of M yr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4156. The GRB sample number is 35.
where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.4763. The GRB sample number is 35. The (−α cpl )-log T 90 -log Age formula is: where T 90 is in unit of s. Age is in unit of Myr. The adjusted R 2 is 0.2569. The GRB sample number is 35. The Mag-log θ j -log M ass formula is: where Mag is in unit of magnitude. θ j is in unit of rad. Mass is in unit of M . The adjusted R 2 is 0.878. The GRB sample number is 35. The log Age-log(1 + z)-log E p,cpl formula is: where Age is in unit of Myr. E p,cpl is in unit of keV. The adjusted R 2 is 0.4777. The GRB sample number is 35.
where Age is in unit of Myr. The adjusted R 2 is 0.4567. The GRB sample number is 35. The log SF R-log D L -log N H formula is: where SFR is in unit of M yr −1 . D L is in unit of 10 28 cm. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.6648. The GRB sample number is 36. The log Γ 0 -log F g -log F Opt11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3764. The GRB sample number is 36. The log Γ 0 -log F pk1 -log F Opt11hr formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3141. The GRB sample number is 36.
The log E p,band -variability 3 -log T 90,i formula is: where E p,band is in unit of keV. T 90,i is in unit of s. The adjusted R 2 is 0.287. The GRB sample number is 36. The log SF R-log(1 + z)-log N H formula is: where SFR is in unit of M yr −1 . N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.6871. The GRB sample number is 36.

47
The log E p,band,i -log L radio,pk -log T 90 formula is: where E p,band,i is in unit of keV. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . T 90 is in unit of s. The adjusted R 2 is 0.4479. The GRB sample number is 36.
The log E iso -log L radio,pk -log E p,band formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.6844. The GRB sample number is 36. The Mag-A V -log E p,cpl,i formula is: where Mag is in unit of magnitude. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2498. The GRB sample number is 37. The log F g -log E p,band -log t radio,pk formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. E p,band is in unit of keV. t radio,pk is in unit of s. The adjusted R 2 is 0.404. The GRB sample number is 37. The log F Opt11hr -log F radio,pk -log T R45,i formula is: where F Opt11hr is in unit of Jy. F radio,pk is in unit of Jy. T R45,i is in unit of s. The adjusted R 2 is 0.4266. The GRB sample number is 37. The log F radio,pk -log HR-log T 50,i formula is: log F radio,pk = (0.34 ± 0.05) × log HR + (0.18 ± 0.023) × log T 50,i + (−3.9 ± 0.029), where F radio,pk is in unit of Jy. T 50,i is in unit of s. The adjusted R 2 is 0.255. The GRB sample number is 37. The log L pk -variability 1 -(−α band ) formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.2743. The GRB sample number is 37. The log F pk3 -log E p,cpl -log t burst formula is: where F pk3 is peak energy flux of 256 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of where T 90 is in unit of s. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Host galaxy offset is in unit of kpc. The adjusted R 2 is 0.6172. The GRB sample number is 38. The log(1 + z)-log P pk4 -log F radio,pk formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F radio,pk is in unit of Jy. The adjusted R 2 is 0.3862. The GRB sample number is 38. The log P pk1 -variability 1 -(−α cpl ) formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.4813. The GRB sample number is 38. The log L pk -variability 2 -(−α band ) formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.2358. The GRB sample number is 38. The log L pk -variability 3 -log θ j formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. θ j is in unit of rad. The adjusted R 2 is 0.3097. The GRB sample number is 38. The log L radio,pk -log T 50 -log L pk formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . T 50 is in unit of s. L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.6609. The GRB sample number is 38. The log t pkOpt -log HR-log Γ 0 formula is: where t pkOpt is in unit of s. The adjusted R 2 is 0.4806. The GRB sample number is 39. The log P pk4 -log t burst -log t pkOpt formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . t burst is in unit of s. t pkOpt is in unit of s. The adjusted R 2 is 0.3097. The GRB sample number is 39. The log F X11hr -variability 2 -log F Opt11hr formula is: where F X11hr is in unit of Jy. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3125. The GRB sample number is 39. The log F g -β X11hr -log Age formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Age is in unit of Myr. The adjusted R 2 is 0.4175. The GRB sample number is 40. The log F Opt11hr -log F pk2 -log T 90,i formula is: where F Opt11hr is in unit of Jy. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . T 90,i is in unit of s. The adjusted R 2 is 0.5628. The GRB sample number is 40. The log Age-log HR-β X11hr formula is: where Age is in unit of Myr. The adjusted R 2 is 0.231. The GRB sample number is 40. The log D L -log θ j -log Age formula is: where D L is in unit of 10 28 cm. θ j is in unit of rad. Age is in unit of Myr. The adjusted R 2 is 0.3565. The GRB sample number is 40. The log F Opt11hr -log θ j -log t pkOpt formula is: where F Opt11hr is in unit of Jy. θ j is in unit of rad. t pkOpt is in unit of s. The adjusted R 2 is 0.2278. The GRB sample number is 40.
The log E p,cpl,i -log F g -log SF R formula is: where E p,cpl,i is in unit of keV. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. SFR is in unit of M yr −1 . The adjusted R 2 is 0.4204. The GRB sample number is 41. The log F g -log F Opt11hr -metallicity formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F Opt11hr is in unit of Jy. Metallicity is the value of 12 where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2218. The GRB sample number is 41. The log t pkOpt,i -log T R45 -log E p,cpl formula is: where t pkOpt,i is in unit of s. T R45 is in unit of s. E p,cpl is in unit of keV. The adjusted R 2 is 0.3051. The GRB sample number is 41. The log SF R-A V -log t burst,i formula is: where SFR is in unit of M yr −1 . t burst,i is in unit of s. The adjusted R 2 is 0.2794. The GRB sample number is 42.
The log L pk -log F g -log Γ 0 formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.6858. The GRB sample number is 42.
The log E p,cpl,i -log F pk1 -log SF R formula is: where E p,cpl,i is in unit of keV. F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . SFR is in unit of M yr −1 . The adjusted R 2 is 0.3181. The GRB sample number is 42.
The log SF R-log L pk -log F Opt11hr formula is: where SFR is in unit of M yr −1 . L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3027. The GRB sample number is 42. The (−β band )-log F g -log N H formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.2912. The GRB sample number is 43. The log F Opt11hr -log F X11hr -log t pkOpt,i formula is: where F Opt11hr is in unit of Jy. F X11hr is in unit of Jy. t pkOpt,i is in unit of s. The adjusted R 2 is 0.3049. The GRB sample number is 43.
where Mag is in unit of magnitude. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4373. The GRB sample number is 44. The Mag-log F Opt11hr -log t burst,i formula is: where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. t burst,i is in unit of s. The adjusted R 2 is 0.4612. The GRB sample number is 44. The Mag-log M ass-log E p,cpl,i formula is: where Mag is in unit of magnitude. Mass is in unit of M . E p,cpl,i is in unit of keV. The adjusted R 2 is 0.8675. The GRB sample number is 44.
The log E p,band,i -log θ j -log M ass formula is: where E p,band,i is in unit of keV. θ j is in unit of rad. Mass is in unit of M . The adjusted R 2 is 0.2478. The GRB sample number is 44. The log Γ 0 -log(1 + z)-log P pk4 formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.6095. The GRB sample number is 44. The Mag-log(1 + z)-log N H formula is: where Mag is in unit of magnitude. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4667. The GRB sample number is 44.
The log E iso -log E p,cpl,i -log t pkOpt,i formula is:

51
where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. E p,cpl,i is in unit of keV. t pkOpt,i is in unit of s. The adjusted R 2 is 0.6507. The GRB sample number is 45. The log t pkOpt -log P pk4 -log E p,cpl formula is: where t pkOpt is in unit of s. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.3112. The GRB sample number is 45. The log D L -log T 50,i -log t radio,pk,i formula is: where D L is in unit of 10 28 cm. T 50,i is in unit of s. t radio,pk,i is in unit of s. The adjusted R 2 is 0.205. The GRB sample number is 45.
where t pkOpt,i is in unit of s. T 90 is in unit of s. The adjusted R 2 is 0.7005. The GRB sample number is 45. The log SF R-log E iso -log F Opt11hr formula is: where SFR is in unit of M yr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3405. The GRB sample number is 46.
The log M ass-metallicity-log SF R formula is: where Mass is in unit of M . Metallicity is the value of 12 + log O/H. SFR is in unit of M yr −1 . The adjusted R 2 is 0.6178. The GRB sample number is 46.
The log E iso -log θ j -log t radio,pk,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. θ j is in unit of rad. t radio,pk,i is in unit of s. The adjusted R 2 is 0.2374. The GRB sample number is 46.
The log E iso -log L radio,pk -log θ j formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . θ j is in unit of rad. The adjusted R 2 is 0.5004. The GRB sample number is 46. The log F Opt11hr -log Age-log T R45,i formula is: where F Opt11hr is in unit of Jy. Age is in unit of Myr. T R45,i is in unit of s. The adjusted R 2 is 0.3872. The GRB sample number is 47. The log P pk4 -log E p,band -log F X11hr formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,band is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.6621. The GRB sample number is 47. The log D L -log HR-log Γ 0 formula is: where D L is in unit of 10 28 cm. The adjusted R 2 is 0.3027. The GRB sample number is 47. The log F g -log P pk1 -log F Opt11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.6153. The GRB sample number is 48. The log HR-variability 3 -(−β band ) formula is: The adjusted R 2 is 0.251. The GRB sample number is 48. The log L radio,pk -log L pk -A V formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . L pk is in unit of 10 52 ergs s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.49. The GRB sample number is 48. The log L pk -log F g -log F radio,pk formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F radio,pk is in unit of Jy. The adjusted R 2 is 0.4055. The GRB sample number is 49. The log L pk -log F g -log t radio,pk,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. t radio,pk,i is in unit of s. The adjusted R 2 is 0.353. The GRB sample number is 49. The log F Opt11hr -log N H -log M ass formula is: where F Opt11hr is in unit of Jy. N H is in unit of 10 21 cm −2 . Mass is in unit of M . The adjusted R 2 is 0.3336. The GRB sample number is 49. The log F g -variability 3 -log T 90,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T 90,i is in unit of s. The adjusted R 2 is 0.4482. The GRB sample number is 49. The log F g -variability 3 -log T R45,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T R45,i is in unit of s. The adjusted R 2 is 0.278. The GRB sample number is 49. The log F g -log(1 + z)-variability 3 formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.3132. The GRB sample number is 49. The log F g -log D L -variability 3 formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. D L is in unit of 10 28 cm. The adjusted R 2 is 0.2954. The GRB sample number is 50. The log HR-log E iso -log F radio,pk formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F radio,pk is in unit of Jy. The adjusted R 2 is 0.2991. The GRB sample number is 50. The log Age-log F Opt11hr -log SSF R formula is: where Age is in unit of Myr. F Opt11hr is in unit of Jy. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.3771. The GRB sample number is 50.
The log E p,band,i -log HR-log θ j formula is: where E p,band,i is in unit of keV. θ j is in unit of rad. The adjusted R 2 is 0.4157. The GRB sample number is 51.
The log SF R-log T 90 -log F Opt11hr formula is: where SFR is in unit of M yr −1 . T 90 is in unit of s. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3356. The GRB sample number is 51. The log L radio,pk -log E iso -log F Opt11hr formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.4846. The GRB sample number is 52. The log F Opt11hr -log Age-log M ass formula is: where F Opt11hr is in unit of Jy. Age is in unit of Myr. Mass is in unit of M . The adjusted R 2 is 0.3177. The GRB sample number is 53. The β X11hr -log E p,band -log F X11hr formula is: where E p,band is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.3324. The GRB sample number is 53.
The log E p,band,i -log HR-log M ass formula is: where E p,band,i is in unit of keV. Mass is in unit of M . The adjusted R 2 is 0.6666. The GRB sample number is 53.
The log Age-log HR-log SSF R formula is: where Age is in unit of Myr. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.2396. The GRB sample number is 53. The Mag-log L pk -log P pk4 formula is: where Mag is in unit of magnitude. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.3453. The GRB sample number is 53. The log M ass-log L pk -metallicity formula is: where Mass is in unit of M . L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.3348. The GRB sample number is 53. The log F Opt11hr -log M ass-log t burst,i formula is: where F Opt11hr is in unit of Jy. Mass is in unit of M . t burst,i is in unit of s. The adjusted R 2 is 0.2911. The GRB sample number is 53. The log Age-log(1 + z)-log F Opt11hr formula is: where Age is in unit of Myr. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2579. The GRB sample number is 53. The log L radio,pk -log E iso -A V formula is: where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.5084. The GRB sample number is 53.
The log M ass-β X11hr -Mag formula is: where Mass is in unit of M . Mag is in unit of magnitude. The adjusted R 2 is 0.8391. The GRB sample number is 54. The log F g -β X11hr -log SSF R formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. log SSF R is in unit of Gyr −1 . The adjusted R 2 is 0.3069. The GRB sample number is 54. The log T 90,i -log E p,band -log F X11hr formula is: where T 90,i is in unit of s. E p,band is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.2015. The GRB sample number is 54. The log F radio,pk -log F Opt11hr -log t radio,pk,i formula is: where F radio,pk is in unit of Jy. F Opt11hr is in unit of Jy. t radio,pk,i is in unit of s. The adjusted R 2 is 0.4246. The GRB sample number is 54. The log F Opt11hr -log F radio,pk -log T 90,i formula is: where F Opt11hr is in unit of Jy. F radio,pk is in unit of Jy. T 90,i is in unit of s. The adjusted R 2 is 0.4369. The GRB sample number is 54.

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The log D L -log F radio,pk -A V formula is: where D L is in unit of 10 28 cm. F radio,pk is in unit of Jy. The adjusted R 2 is 0.237. The GRB sample number is 54. The log M ass-log T 50 -log SF R formula is: where Mass is in unit of M . T 50 is in unit of s. SFR is in unit of M yr −1 . The adjusted R 2 is 0.6319. The GRB sample number is 54. The log F Opt11hr -log(1 + z)-log F radio,pk formula is: where F Opt11hr is in unit of Jy. F radio,pk is in unit of Jy. The adjusted R 2 is 0.4305. The GRB sample number is 54.
The log t pkOpt -log F g -log F Opt11hr formula is: where F Opt11hr is in unit of Jy. t pkOpt is in unit of s. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.309. The GRB sample number is 55.
The log SF R-metallicity-A V formula is: where SFR is in unit of M yr −1 . Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.265. The GRB sample number is 55. The Mag-log M ass-log t burst,i formula is: where Mag is in unit of magnitude. Mass is in unit of M . t burst,i is in unit of s. The adjusted R 2 is 0.8658. The GRB sample number is 57. The Mag-log T 90 -log t burst,i formula is: where Mag is in unit of magnitude. T 90 is in unit of s. t burst,i is in unit of s. The adjusted R 2 is 0.2815. The GRB sample number is 57. The log HR-log θ j -log E p,cpl,i formula is: where θ j is in unit of rad. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.605. The GRB sample number is 57. The log L pk -log θ j -log t burst,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. θ j is in unit of rad. t burst,i is in unit of s. The adjusted R 2 is 0.2276. The GRB sample number is 57. The log F g -(−β band )-β X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.4589. The GRB sample number is 58.
The log SF R-log E iso -metallicity formula is: where SFR is in unit of M yr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.3087. The GRB sample number is 58. The log F g -log E p,band -β X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. E p,band is in unit of keV. The adjusted R 2 is 0.6254. The GRB sample number is 58.
The log E p,band -variability 3 -log F g formula is: where E p,band is in unit of keV. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.3281. The GRB sample number is 58. The log SF R-log D L -metallicity formula is: where SFR is in unit of M yr −1 . D L is in unit of 10 28 cm. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.584. The GRB sample number is 59. The log F pk1 -log E p,band -β X11hr formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.5634. The GRB sample number is 59. The log L pk -Mag-log T 50,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Mag is in unit of magnitude. T 50,i is in unit of s. The adjusted R 2 is 0.3217. The GRB sample number is 59. The log SF R-log(1 + z)-metallicity formula is: where SFR is in unit of M yr −1 . Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.6034. The GRB sample number is 59. The log F g -log L pk -metallicity formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.2819. The GRB sample number is 61.
The log E iso -log L radio,pk -log t radio,pk formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . t radio,pk is in unit of s. The adjusted R 2 is 0.4832. The GRB sample number is 61. The log L radio,pk -log E iso -log T 90,i formula is: Statistics on GRBs 57 where L radio,pk is in rest-frame 8.46 GHz, the unit is 10 40 erg s −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. T 90,i is in unit of s. The adjusted R 2 is 0.4534. The GRB sample number is 61. The log M ass-A V -log E p,cpl,i formula is: where Mass is in unit of M . E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2425. The GRB sample number is 62.
where Mag is in unit of magnitude. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.291. The GRB sample number is 62. The log D L -log F radio,pk -log t radio,pk,i formula is: where D L is in unit of 10 28 cm. F radio,pk is in unit of Jy. t radio,pk,i is in unit of s. The adjusted R 2 is 0.242. The GRB sample number is 62. The log L pk -Mag-log T 90,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Mag is in unit of magnitude. T 90,i is in unit of s. The adjusted R 2 is 0.3448. The GRB sample number is 62. The log SF R-log E iso -log M ass formula is: where SFR is in unit of M yr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.5535. The GRB sample number is 63. The log M ass-log E iso -metallicity formula is: where Mass is in unit of M . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.3948. The GRB sample number is 63. The log SSF R-log F g -log Age formula is: where log SSF R is in unit of Gyr −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Age is in unit of Myr. The adjusted R 2 is 0.2893. The GRB sample number is 63. The log θ j -log T 90 -log t burst,i formula is: where θ j is in unit of rad. T 90 is in unit of s. t burst,i is in unit of s. The adjusted R 2 is 0.2592. The GRB sample number is 63. The log D L -log θ j -log E p,cpl,i formula is: where D L is in unit of 10 28 cm. θ j is in unit of rad. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.299. The GRB sample number is 63. The log D L -log θ j -log t burst,i formula is: where D L is in unit of 10 28 cm. θ j is in unit of rad. t burst,i is in unit of s. The adjusted R 2 is 0.2289. The GRB sample number is 63. The log M ass-log D L -metallicity formula is: where Mass is in unit of M . D L is in unit of 10 28 cm. Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.4144. The GRB sample number is 64. The Mag-log F Opt11hr -log M ass formula is: where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. Mass is in unit of M . The adjusted R 2 is 0.8621. The GRB sample number is 64. The log(1 + z)-log θ j -log N H formula is: where θ j is in unit of rad. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.3328. The GRB sample number is 64.
where D L is in unit of 10 28 cm. θ j is in unit of rad. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.3219. The GRB sample number is 64.
The log M ass-log(1 + z)-metallicity formula is: where Mass is in unit of M . Metallicity is the value of 12 + log O/H. The adjusted R 2 is 0.4282. The GRB sample number is 64. The Mag-log(1 + z)-log F Opt11hr formula is: where Mag is in unit of magnitude. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3878. The GRB sample number is 64. The log F g -(−β band )-log F X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F X11hr is in unit of Jy. The adjusted R 2 is 0.4304. The GRB sample number is 65. The log F g -log E p,band -log F X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. E p,band is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.8306. The GRB sample number is 65. The log F pk1 -log E p,band -log F X11hr formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,band is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.7179. The GRB sample number is 66.
The log E p,band,i -log F g -log θ j formula is:

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where E p,band,i is in unit of keV. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. θ j is in unit of rad. The adjusted R 2 is 0.223. The GRB sample number is 66. The log P pk4 -log E p,band -log F Opt11hr formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,band is in unit of keV. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.4235. The GRB sample number is 67. The log Age-log(1 + z)-log HR formula is: where Age is in unit of Myr. The adjusted R 2 is 0.3026. The GRB sample number is 67. The log SSF R-log Age-log M ass formula is: where log SSF R is in unit of Gyr −1 . Age is in unit of Myr. Mass is in unit of M . The adjusted R 2 is 0.2948. The GRB sample number is 68. The log SSF R-log E iso -log Age formula is: where log SSF R is in unit of Gyr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Age is in unit of Myr. The adjusted R 2 is 0.2254. The GRB sample number is 68.
The log E p,band,i -log F g -log M ass formula is: where E p,band,i is in unit of keV. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.2071. The GRB sample number is 68. The log SF R-log D L -log M ass formula is: where SFR is in unit of M yr −1 . D L is in unit of 10 28 cm. Mass is in unit of M . The adjusted R 2 is 0.6629. The GRB sample number is 69. The log T 90 -log θ j -log E p,band,i formula is: where T 90 is in unit of s. θ j is in unit of rad. E p,band,i is in unit of keV. The adjusted R 2 is 0.2107. The GRB sample number is 69. The log SF R-log(1 + z)-log M ass formula is: where SFR is in unit of M yr −1 . Mass is in unit of M . The adjusted R 2 is 0.687. The GRB sample number is 69. The Mag-log(1 + z)-A V formula is: where Mag is in unit of magnitude. The adjusted R 2 is 0.408. The GRB sample number is 69. The log t burst,i -log L pk -log F Opt11hr formula is: log t burst,i = (−0.18 ± 0.021) × log L pk + (0.21 ± 0.047) × log F Opt11hr + (3.4 ± 0.24), where t burst,i is in unit of s. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2132. The GRB sample number is 70. The log F X11hr -log F g -log θ j formula is: log F X11hr = (0.43 ± 0.039) × log F g + (0.51 ± 0.061) × log θ j + (−6.9 ± 0.089), where F X11hr is in unit of Jy. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. θ j is in unit of rad. The adjusted R 2 is 0.3077. The GRB sample number is 71.
The log M ass-log P pk4 -Mag formula is: log M ass = (−0.18 ± 0.09) × log P pk4 + (−0.14 ± 0.023) × M ag + (6.8 ± 0.5), where Mass is in unit of M . P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . Mag is in unit of magnitude. The adjusted R 2 is 0.8775. The GRB sample number is 72. The log F pk2 -(−β band )-log E p,band,i formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,band,i is in unit of keV. The adjusted R 2 is 0.2559. The GRB sample number is 73.
The log M ass-log HR-Mag formula is: log M ass = (−0.33 ± 0.14) × log HR + (−0.12 ± 0.025) × M ag + (7.1 ± 0.54), where Mass is in unit of M . Mag is in unit of magnitude. The adjusted R 2 is 0.8604. The GRB sample number is 73.
The log E iso -log P pk1 -log E p,band,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,band,i is in unit of keV. The adjusted R 2 is 0.2987. The GRB sample number is 73. The log F pk2 -log(1 + z)-(−β band ) formula is: where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.2389. The GRB sample number is 73. The log M ass-(−β band )-log E p,band,i formula is: where Mass is in unit of M . E p,band,i is in unit of keV. The adjusted R 2 is 0.2063. The GRB sample number is 74.
The log E p,band,i -log D L -log M ass formula is: where E p,band,i is in unit of keV. D L is in unit of 10 28 cm. Mass is in unit of M . The adjusted R 2 is 0.2109. The GRB sample number is 74. The log T 50,i -(−α cpl )-log t burst,i formula is: log T 50,i = (0.22 ± 0.072) × (−α cpl ) + (0.33 ± 0.02) × log t burst,i + (−0.16 ± 0.11), where T 50,i is in unit of s. t burst,i is in unit of s. The adjusted R 2 is 0.2056. The GRB sample number is 75.

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The log F pk1 -log HR-log SF R formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . SFR is in unit of M yr −1 . The adjusted R 2 is 0.2425. The GRB sample number is 76. The log L pk -log θ j -log M ass formula is: log L pk = (−1.5 ± 0.1) × log θ j + (0.32 ± 0.06) × log M ass + (−4.9 ± 0.56), where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. θ j is in unit of rad. Mass is in unit of M . The adjusted R 2 is 0.2767. The GRB sample number is 76. The log θ j -log(1 + z)-log F X11hr formula is: where θ j is in unit of rad. F X11hr is in unit of Jy. The adjusted R 2 is 0.2238. The GRB sample number is 76. The log D L -log F g -log Age formula is: where D L is in unit of 10 28 cm. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Age is in unit of Myr. The adjusted R 2 is 0.3209. The GRB sample number is 77. The log F g -log P pk1 -A V formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.4258. The GRB sample number is 77. The log P pk3 -log(1 + z)-(−β band ) formula is: log P pk3 = (−0.82 ± 0.067) × log(1 + z) + (0.33 ± 0.14) × (−β band ) + (0.64 ± 0.33), where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2232. The GRB sample number is 77. The log T R45,i -log F g -log t pkOpt formula is: where T R45,i is in unit of s. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. t pkOpt is in unit of s. The adjusted R 2 is 0.3476. The GRB sample number is 78. The log P pk4 -log F g -log SF R formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. SFR is in unit of M yr −1 . The adjusted R 2 is 0.3386. The GRB sample number is 79. The log SF R-log E iso -A V formula is: where SFR is in unit of M yr −1 . E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.3195. The GRB sample number is 81. The log P pk4 -log L pk -(−β band ) formula is: log P pk4 = (0.28 ± 0.04) × log L pk + (0.18 ± 0.14) × (−β band ) + (0.39 ± 0.32), where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.2003. The GRB sample number is 81. The log D L -log θ j -log M ass formula is: where D L is in unit of 10 28 cm. θ j is in unit of rad. Mass is in unit of M . The adjusted R 2 is 0.3388. The GRB sample number is 81. The log F Opt11hr -β X11hr -log M ass formula is: where F Opt11hr is in unit of Jy. Mass is in unit of M . The adjusted R 2 is 0.252. The GRB sample number is 82.
The log M ass-log T 90 -Mag formula is: where Mass is in unit of M . T 90 is in unit of s. Mag is in unit of magnitude. The adjusted R 2 is 0.8661. The GRB sample number is 82. The log t burst,i -log P pk4 -log F Opt11hr formula is: where t burst,i is in unit of s. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2451. The GRB sample number is 83. The log P pk4 -log D L -log t pkOpt formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . D L is in unit of 10 28 cm. t pkOpt is in unit of s. The adjusted R 2 is 0.38. The GRB sample number is 84. The log F Opt11hr -log F X11hr -log M ass formula is: where F Opt11hr is in unit of Jy. F X11hr is in unit of Jy. Mass is in unit of M . The adjusted R 2 is 0.2405. The GRB sample number is 88. The log SF R-log D L -A V formula is: where SFR is in unit of M yr −1 . D L is in unit of 10 28 cm. The adjusted R 2 is 0.4998. The GRB sample number is 90. The log SF R-log(1 + z)-A V formula is: where SFR is in unit of M yr −1 . The adjusted R 2 is 0.5415. The GRB sample number is 90. The log HR-log T 50 -log E p,band,i formula is: where T 50 is in unit of s. E p,band,i is in unit of keV. The adjusted R 2 is 0.243. The GRB sample number is 96. The log F g -β X11hr -log M ass formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.2965. The GRB sample number is 97.
The log E iso -log F X11hr -log E p,cpl,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F X11hr is in unit of Jy. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.386. The GRB sample number is 97. The log P pk4 -log F g -log N H formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.4773. The GRB sample number is 104.
The log M ass-log F Opt11hr -A V formula is: where Mass is in unit of M . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2312. The GRB sample number is 104.
The log F g -log L pk -log N H formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.2658. The GRB sample number is 104. The log F Opt11hr -log M ass-log T 50,i formula is: where F Opt11hr is in unit of Jy. Mass is in unit of M . T 50,i is in unit of s. The adjusted R 2 is 0.2729. The GRB sample number is 105. The log F pk1 -(−α cpl )-β X11hr formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.2512. The GRB sample number is 108. The log F g -log E p,band -log F Opt11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. E p,band is in unit of keV. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.6036. The GRB sample number is 108. The log F pk1 -log E p,cpl -β X11hr formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.4823. The GRB sample number is 108. The log M ass-log L pk -A V formula is: log M ass = (0.24 ± 0.031) × log L pk + (0.3 ± 0.068) × A V + (9.3 ± 0.065), where Mass is in unit of M . L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.208. The GRB sample number is 110.
The log P pk3 -log(1 + z)-log F g formula is: log P pk3 = (−0.73 ± 0.057) × log(1 + z) + (0.47 ± 0.011) × log F g + (0.74 ± 0.033), where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.6368. The GRB sample number is 112. The log F Opt11hr -log M ass-log T R45,i formula is: where F Opt11hr is in unit of Jy. Mass is in unit of M . T R45,i is in unit of s. The adjusted R 2 is 0.2692. The GRB sample number is 115. The log F g -(−β band )-log T R45,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T R45,i is in unit of s. The adjusted R 2 is 0.3425. The GRB sample number is 117. The log F g -log E p,band -log T R45,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. E p,band is in unit of keV. T R45,i is in unit of s. The adjusted R 2 is 0.6001. The GRB sample number is 117.
The log E p,band -log T R45 -log P pk4 formula is: where E p,band is in unit of keV. T R45 is in unit of s. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2789. The GRB sample number is 118. The log F Opt11hr -log F g -log M ass formula is: log F Opt11hr = (0.33 ± 0.055) × log F g + (−0.34 ± 0.063) × log M ass + (−2 ± 0.6), where F Opt11hr is in unit of Jy. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.2536. The GRB sample number is 120.
The log E iso -log F X11hr -β X11hr formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F X11hr is in unit of Jy. The adjusted R 2 is 0.2423. The GRB sample number is 121. The β X11hr -log P pk4 -log T R45,i formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . T R45,i is in unit of s. The adjusted R 2 is 0.3002. The GRB sample number is 123. The log T 90 -log E iso -log N H formula is: where T 90 is in unit of s. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. N H is in unit of 10 21 cm −2 . The adjusted R 2 is 0.2526. The GRB sample number is 125. The log M ass-log D L -log F Opt11hr formula is: where Mass is in unit of M . D L is in unit of 10 28 cm. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2064. The GRB sample number is 126. The log D L -log θ j -A V formula is: where D L is in unit of 10 28 cm. θ j is in unit of rad. The adjusted R 2 is 0.2105. The GRB sample number is 126. The log M ass-log(1 + z)-log F Opt11hr formula is: where Mass is in unit of M . F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2156. The GRB sample number is 126.
The log F g -log L pk -(−β band ) formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.2296. The GRB sample number is 127.
The log E p,band,i -log T 90 -log L pk formula is: where E p,band,i is in unit of keV. T 90 is in unit of s. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. The adjusted R 2 is 0.4998. The GRB sample number is 127. The log F X11hr -β X11hr -log T 50,i formula is: where F X11hr is in unit of Jy. T 50,i is in unit of s. The adjusted R 2 is 0.2084. The GRB sample number is 129. The log P pk4 -log D L -β X11hr formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . D L is in unit of 10 28 cm. The adjusted R 2 is 0.3358. The GRB sample number is 130. The log T 50,i -log E iso -log M ass formula is: log T 50,i = (0.29 ± 0.0082) × log E iso + (−0.15 ± 0.022) × log M ass + (2.1 ± 0.21), where T 50,i is in unit of s. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.2199. The GRB sample number is 130. The log F g -log L pk -log M ass formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.3385. The GRB sample number is 130. The log P pk4 -log(1 + z)-β X11hr formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.3504. The GRB sample number is 130. The log T 90 -variability 1 -log E iso formula is: where T 90 is in unit of s. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. The adjusted R 2 is 0.2188. The GRB sample number is 132. The log F X11hr -β X11hr -log T 90,i formula is: where F X11hr is in unit of Jy. T 90,i is in unit of s. The adjusted R 2 is 0.2234. The GRB sample number is 133. The (−α band )-log HR-log T 90,i formula is: where T 90,i is in unit of s. The adjusted R 2 is 0.2783. The GRB sample number is 135. The log HR-(−α band )-log E p,band,i formula is: where E p,band,i is in unit of keV. The adjusted R 2 is 0.442. The GRB sample number is 136. The log L pk -log P pk4 -log E p,cpl,i formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2905. The GRB sample number is 136. The log F g -log L pk -log E p,cpl,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2238. The GRB sample number is 139. The log L pk -log F g -log t burst,i formula is: log L pk = (0.57 ± 0.025) × log F g + (−0.46 ± 0.027) × log t burst,i + (0.58 ± 0.062), where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. t burst,i is in unit of s. The adjusted R 2 is 0.2877. The GRB sample number is 140.
The log E iso -log T R45 -log t burst,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. T R45 is in unit of s. t burst,i is in unit of s. The adjusted R 2 is 0.2667. The GRB sample number is 140. The log M ass-log(1 + z)-A V formula is: where Mass is in unit of M . The adjusted R 2 is 0.2586. The GRB sample number is 146.
The log E iso -log T 90 -β X11hr formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. T 90 is in unit of s. The adjusted R 2 is 0.516. The GRB sample number is 149. The log P pk4 -log E p,cpl -log F X11hr formula is: log P pk4 = (0.18 ± 0.05) × log E p,cpl + (0.22 ± 0.013) × log F X11hr + (1.7 ± 0.14), where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,cpl is in unit of keV. F X11hr is in unit of Jy. The adjusted R 2 is 0.2295. The GRB sample number is 152. The log F X11hr -log E iso -log F Opt11hr formula is: where F X11hr is in unit of Jy. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.2229. The GRB sample number is 153.
The log E iso -log T 50 -log F X11hr formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. T 50 is in unit of s. F X11hr is in unit of Jy. The adjusted R 2 is 0.3408. The GRB sample number is 153. The log F g -log(1 + z)-β X11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.2953. The GRB sample number is 153. The log F pk1 -log HR-log M ass formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 .
Mass is in unit of M . The adjusted R 2 is 0.3515. The GRB sample number is 157.
The log E iso -log F pk1 -log E p,band,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,band,i is in unit of keV. The adjusted R 2 is 0.564. The GRB sample number is 158.
The log E iso -log T 90 -log F X11hr formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. T 90 is in unit of s. F X11hr is in unit of Jy. The adjusted R 2 is 0.3803. The GRB sample number is 159. The log F g -log L pk -log F Opt11hr formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3953. The GRB sample number is 171. The log T 90 -log E iso -log M ass formula is: where T 90 is in unit of s. E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. Mass is in unit of M . The adjusted R 2 is 0.3543. The GRB sample number is 172. The log P pk4 -log F g -log t burst,i formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. t burst,i is in unit of s. The adjusted R 2 is 0.3963. The GRB sample number is 172.
The log P pk4 -log HR-log F X11hr formula is: log P pk4 = (0.28 ± 0.047) × log HR + (0.23 ± 0.016) × log F X11hr + (2.2 ± 0.12), where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F X11hr is in unit of Jy. The adjusted R 2 is 0.2558. The GRB sample number is 182. The log F g -log T 50,i -log E p,cpl,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T 50,i is in unit of s. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.4764. The GRB sample number is 185. The log P pk4 -log T 90 -log F X11hr formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . T 90 is in unit of s. F X11hr is in unit of Jy. The adjusted R 2 is 0.2226. The GRB sample number is 216. The log F g -log L pk -log T 50,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. T 50,i is in unit of s. The adjusted R 2 is 0.4039. The GRB sample number is 218. The log HR-variability 1 -(−β band ) formula is: the adjusted R 2 is 0.2164. The GRB sample number is 219. The log L pk -log F g -A V formula is: where L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.2064. The GRB sample number is 220. The log F g -variability 1 -log P pk1 formula is: log F g = (−1.9 ± 0.58) × variability 1 + (0.83 ± 0.038) × log P pk1 + (0.41 ± 0.055), where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.4694. The GRB sample number is 227.
The log E iso -(−α cpl )-log E p,cpl,i formula is: where E iso is in unit of 10 52 ergs and in rest-frame 1-10 4 keV energy band. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2692. The GRB sample number is 228. The log F X11hr -log F g -log F Opt11hr formula is: where F X11hr is in unit of Jy. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. F Opt11hr is in unit of Jy. The adjusted R 2 is 0.3134. The GRB sample number is 232. The log F g -log D L -log E p,cpl formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. D L is in unit of 10 28 cm. E p,cpl is in unit of keV. The adjusted R 2 is 0.2516. The GRB sample number is 248. The log D L -log T 90 -log E p,cpl,i formula is: where D L is in unit of 10 28 cm. T 90 is in unit of s. E p,cpl,i is in unit of keV. The adjusted R 2 is 0.2192. The GRB sample number is 257. The log F g -log L pk -log T 90,i formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. L pk is in unit of 10 52 erg s −1 , and in 1-10 4 keV energy band. T 90,i is in unit of s. The adjusted R 2 is 0.4429. The GRB sample number is 312. The log F g -log T R45 -log E p,cpl formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. T R45 is in unit of s. E p,cpl is in unit of keV. The adjusted R 2 is 0.5627. The GRB sample number is 362. The log P pk4 -log D L -log F g formula is: where P pk4 is peak photon flux of 1 s time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . D L is in unit of 10 28 cm. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.3421. The GRB sample number is 419. The log HR-log F pk2 -(−α spl ) formula is: log HR = (0.17 ± 0.036) × log F pk2 + (−0.9 ± 0.039) × (−α spl ) + (2.2 ± 0.075), where F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.5718. The GRB sample number is 490. The log HR-log T 50 -(−α spl ) formula is: where T 50 is in unit of s. The adjusted R 2 is 0.6123. The GRB sample number is 497. The log HR-log F pk1 -log E p,cpl formula is: where F pk1 is peak energy flux of 1 s time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,cpl is in unit of keV. The adjusted R 2 is 0.8214. The GRB sample number is 713. The log P pk3 -log F g -(−α cpl ) formula is: where P pk3 is peak photon flux of 1024 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.4674. The GRB sample number is 913. The log HR-log F pk4 -(−α cpl ) formula is: log HR = (0.42 ± 0.024) × log F pk4 + (−0.31 ± 0.025) × (−α cpl ) + (0.96 ± 0.026), where F pk4 is peak energy flux of 1024 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.3125. The GRB sample number is 937.
The log HR-log P pk1 -(−β band ) formula is: where P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . The adjusted R 2 is 0.2471. The GRB sample number is 1092. The log F g -log P pk1 -log E p,band formula is: where F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. P pk1 is peak photon flux of 64 ms time bin in 10-1000 keV, and in unit of photons cm −2 s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.4116. The GRB sample number is 1121.
The log E p,band -log HR-log F pk2 formula is: where E p,band is in unit of keV. F pk2 is peak energy flux of 64 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . The adjusted R 2 is 0.6307. The GRB sample number is 1130. The log T 50 -log HR-(−α cpl ) formula is: where T 50 is in unit of s. The adjusted R 2 is 0.2224. The GRB sample number is 1140.
The log HR-log T 50 -(−β band ) formula is: where T 50 is in unit of s. The adjusted R 2 is 0.201. The GRB sample number is 1183. The log F pk4 -(−α band )-log E p,band formula is: where F pk4 is peak energy flux of 1024 ms time bin in rest-frame 1-10 4 keV energy band, and in unit of 10 −6 ergs cm −2 s −1 . E p,band is in unit of keV. The adjusted R 2 is 0.3073. The GRB sample number is 1192. The log T 90 -log F g -(−α cpl ) formula is: log T 90 = (0.56 ± 0.013) × log F g + (0.32 ± 0.03) × (−α cpl ) + (0.6 ± 0.032), where T 90 is in unit of s. F g is in unit of 10 −6 ergs cm −2 , and in 20-2000 keV energy band. The adjusted R 2 is 0.3263. The GRB sample number is 1347.

DISCUSSIONS
There are also many other interesting quantities which are not listed in this data sample, mainly because the quantities are limited to a few GRBs, or they are not quite well defined or not widely accepted. People may want to use the data shown here as an reservoir, and add any other data interested to perform the statistical studies. With Fourier transformation of the prompt light curves, the slope in the frequency domain. Properties of the precursors are not included neither, though they are thought no much different from the prompt emission themselves (Burlon et al. 2008(Burlon et al. , 2009). Because of the universal behavior of the afterglow light curves (Zhang et al. 2006a), there are quite a few parameters including the temporal decay index, the ending time, and spectral index in each phase (they are in X-ray, and mainly contains only one segment, i.e., no breaks in the spectra), while the phases include the steep decay phase, the plateau phase, the normal decay phase and the jet break phase. There are also parameters for the X-ray flares in the afterglow, such as the number of the flares, typical duration, rising temporal index, decay temporal index, spectrum index and luminosities. Beside the spectral indices, there are also plenty of detailed spectral lines from the afterglows and from the host galaxies, which can be used as characteristic quantities. One can use those data and the data listed in this paper for a combined analysis. On the other hand, the combination of the parameters can also be taken as independent parameters, such as the average luminosity L iso ≡ (1+z)Eiso

T90
, and the spectral index difference of the Band spectrum α − β. For the detailed study on special quantities, one can dig more data from the original data sets. For example, P pk and F pk have four time bins mainly because different authors are interested in different time bins. It is not proper to simply take them equal, and four different values are gathered in this work. In most cases, for each GRB, only a few time bins are available. One could produce the P pk for 1024 ms bin (for example) if the original light curves are available. Therefore, digging into the original data from the satellites will provide much more extra data than them shown in this work. On the other hand, the light curves in different bands are also interesting, of which the corresponding quantities are not shown here. One could expand the parameters by digging into the raw data.
One should be careful of using these data because of the variety sources of the data. With different instruments, the energy band is different and the sensitivity is different. For example, the T 90 is energy dependent, therefore, T 90 from different instruments should not be taken as the same parameter if the energy band for different instruments differs too much. A proper way to handle this selection effect is to convert the light curves in the same energy band, which needs the spectrum models. However, in many cases, the spectrum model for each GRB is not the same, and for some GRBs, it is possible that two or more models are the same goodness for the spectrum fitting. Therefore, it is model dependent. For the complexity and the model dependent problem, we only include the direct available data, while leaving these converting works in the future. The same problem also apply to T 50 , spectral lag, HR, fluence F , photon flux P , luminosity L and isotropic equivalent energy E iso . To avoid these problems, we have converted them into the same band, either in a mostly used energy band, or a full band (like 1 − 10, 000keV), by employing the spectral fitting parameters. However, these conversions are model depended as for different bursts, the spectral fitting model might be different, and the real model might not be the best fitting model. Even the data can be converted into the values in the same band, there is still cosmological effects. The same band is not the same band in the rest frame. Because of the cosmological redshift, in order to compare in the same band, one should convert the quantities in the same band in the rest frame. However, even in the same band in the rest frame, the cosmological evolution effect is not removable. It is still unclear on how the GRBs evolves with the universe evolution. On the other hand, with plenty of the data at different redshift, it is possible to study the evolution.
To avoid the problems arisen by the selection effects, we may mainly concentrate on the physical quantities (like luminosities, energies) rather than the observational values (like fluence, photon flux), and we may use the quantities in the rest frame rather than in the observer-frame. However, they both rely on the redshift detection, which is hard and consequently the number of the sample is much smaller. As a balance, one may use the data as many as possible, while keep in mind about the reliability of the GRBs without redshift obtained. On the other hand, the relations between observational quantities may reveal which is the selection effect. If one finds the properties only shown in the brighter GRBs (more photons observed), those may probably be not the intrinsic connections.
Even with the selection effect considered, one may see from the figures shown in the figure sets, the correlations are not tight. Though they are all consistent with the previously found correlations, such as the Amati relation  ) provided a striking clue for ultra-long GRB 111209A, which is driven by spin-down radiation from a highly magnetized millisecond pulsar, known as a magnetar. Gompertz & Fruchter (2017) made efforts to place constraints on the magnetar model. Based on spectrum (or hardness ratio), there are soft GRBs, hard GRBs, and some are very soft and being classified into X-ray rich GRBs or even X-ray flashes. Based on the luminosity, there are high luminosity GRBs and low luminosity GRBs. Based on the connection with other phenomena, there are SN Ic connected GRBs, no SN connected GRBs into a very dim flux, kilonova connected GRBs and strong GeV connected GRBs. From the physical origin view, there are also different subgroups. Based on the progenitors, they can be massive stars, BH-NS binaries, NS-NS binaries, and even BH-WD binaries and NS-WD binaries. Based on the central engine, they can be BZ mechanism dominated, NDAF dominated and magentar dominated. Based on the radiation mechanism, they can be synchrotron radiation dominated, inverse Compton scattering dominated or photosphere emission dominated.
Therefore, the direct analysis may not reveal the underline pattern, mainly because of the selection effect and the clustering effect. For the selection effect, one should try to figure out each factor, if possible. For the clustering effect, one should try to find the subgroups from the full sample. A widely known clustering is the long-soft GRBs and the short-hard GRBs. However, it is very likely the GRBs should be classified into more subgroups. From the point of view of the central engine, there might be BH accretion disk system powered or neutron star powered. From the view of the progenitors, there might be massive star collapsar, BH-WD merger, BH-star merger for LGRBs, while BH-NS merger, NS-NS merger for SGRBs. The other possibilities, such as NS-star merger and NS-WD merger, might also be hidden in the sample. From the view of the radiation mechanism, there might be photosphere emission, synchrotron radiation, inverse Compton scattering (depending on the origins of the seed photons and the electrons/protons, much more subgroups could be needed) etc. From point of the view of the ejecta, it could be higher relativistic or mildly relativistic. From the view of the environment, the number density could be uniformly distributed or wind like distribution, and it could be a dense environment or a thin one. For the optical counterpart, it could be supernova connected, kilonova connected, no supernova connected, or even a dim optical counterpart. From the view of the host galaxy, there might be a spiral galaxy, a field galaxy, or other types. More detailed or other classification criteria can be proposed. The different origins or mechanisms represent in the properties of the data, and should be revealed from the detailed and comprehensive study on the data.
The clustering and the correlation analysis actually affect each other. With a proper classification, such as if the NS-WD merger GRBs are gathered by some properties, the correlation between two or three parameters might be much more tighter, and the correlation might be more useful for a standard candle relation, or for a redshift indicator etc. On the other hand, the correlation can be used as indicator for clustering. For example, the E p − E iso relation is often used as one of the indicator for a certain GRB being a long or short GRB. The GRB lies on the Amati relation is more likely a collapsar originated GRB, while the outlier is more likely a merger originated GRB. Therefore, with more detailed clustering, the correlation inside a certain group of GRBs is more reliable and reveals deeper of the nature, which need more and more data accumulation. Once the pure subset has been found, it could be used as a standard candle, and could be an ideal tool for the cosmology into high redshift region.
For the clustering, an example is the classification of the hard SGRBs and the soft LGRBs. There are also other independent classifications, like the ultra-long GRBs, low luminosity GRBs. These are classified by one or two parameters of the GRBs. Machine learning is a set of promising methods, such as the k-means, SVC, Principal Component Analysis (Zhou 2016). They are able to in a much higher dimension, and in high nonlinear manners. Similar as has been done in other areas of astrophysics (e.g. Elorrieta et al. 2016). However, the machine learning methods are often like black boxes. One is easy to get the result of the clustering, but is difficult to find the criteria of the clustering. One of the main aims to figure out the black box, which reveals the intrinsic properties, similar as the clustering process shown in Zhang et al. (2009b). Except for clustering, machine learning can also for parameter predicting. For example, to predict the redshift, similar as being done to the SDSS galaxies (Hoyle et al. 2015). As the lack of the massive data as used in general usage of machine learning, deep learning may also be a promising approach.
High energy radiation in GeV band has been detected in quite a few Fermi GRBs. From the binned light curves, they are likely from the afterglow . However, there are many sources may produce GeV photons, such as, the synchrotron and the synchrotron self inverse Compton scattering (SSC) emission of the long-lasting forward external shock can contribute; the GRB central engines are continually active and the SSC emission of the continued internal shocks may produce GeV photons; in a relativistic reverse shock formed, the prompt optical/X-ray/γ-ray photons are inversely Compton scattered by the accelerated electrons in the forward shock region; and the external inverse Compton scattering (EIC) in the late-afterglow phase caused by X-ray flares may also give rise to GeV emission (Zou et al. 2009). It is still not clear about which part is dominated in the high energy radiation. With the date base of the information in other aspects, the high energy radiation may be used to compare with the data base, and it may reveal some connections. One may use these connections to figure out the origin of the high energy GeV emission, while the high energy data may come from all of the related instruments including Gravitational wave burst is accompanied with the SGRBs, as they are believed from double compact object (NS-NS or NS-BH). It has been identified the NS-NS origin since the detection of GW170817/GRB 170817A (Abbott et al. 2017d,e). However, the observed GRB 170817A was a weak SGRB, which is much different from the normal SGRBs. As the distance of GRB 170817A is much closer than the other SGRBs, the birth rate is consequently much higher than the others, if we consider they belong to two different subgroups. It is possible that NS-NS mergers may produce weak SGRBs, whereas BH-NS mergers may produce strong SGRBs, which are the mostly observed normal SGRBs. There are other gravitational wave bursts have been observed, such as GW150914 (Abbott et al. 2016c). They have confirmed the presence of a BH that is tens of solar masses (for GW150914, it was the merger of 36 +5 −4 M BH and 29 +4 −4 M BH (Abbott et al. 2016c)). Therefore, comparing with NS-NS merger, which is the merger of several M object and several M object, the BH-NS merger is the merger of tens of M object and several M object. The BH-NS merger might be naturally stronger and consequently produces the strong SGRBs. If the hypothesis is correct, these two different objects should have different properties on other aspects. In the hundreds of the observed SGRBs, these two origins should both been included. Considering the operation of the LIGO was just a few months, an NS-NS GRB (GRB 170817A) was observed. It indicates there are much more this kind of GRBs have been observed and been archived when LIGO was not in operation. With the comprehensive table of different aspects of all the GRBs, one may try to figure out the subgroups in the SGRBs. The NS-NS and NS-BH origin may be distinguished, and maybe more subgroups of other origins can be uncovered. Until now, the BH-NS merger has not been confirmed by the gravitational wave detector. With the future operation with longer time, and with the enhanced instruments, the BH-NS merger should be identified.

CONCLUSION
In this work, we collected a large number of data for 6289 gamma-ray bursts (GRBs) from different literature, GCN (Gamma-ray Coordinates Network), website data bases and the calculations. The data include four parts: basic information, prompt emission, afterglow and host galaxy (in total 46 items for each GRB). With this complete table, we performed a comprehensive statistical study. When we did statistics, we also changed some parameters from observer-frame to rest-frame, we use label "i" to remark. For example, the duration of 5% to 95% γ-ray fluence (T 90 ) is in observer-frame, T 90,i is T 90 in rest-frame.This work includes the following 6 items: (1) We imputed the missing errors with multiple imputation (Rubin 1987(Rubin , 1996 by chained equations (MICE); (2) We calculated a small part of peak energy flux (F pk ) in rest-frame 1-10 4 keV energy band, peak photon flux (P pk ) in observerframe 10-1000 keV energy band, fluence (F g ) in observer-frame 20-2000 keV energy band, hardness ratio (HR) between observer-frame 100-2000 and 20-100 keV energy band, isotropic γ-ray energy (E iso ) in rest-frame 1 − 10 4 keV energy band and peak luminosity (L pk ) in rest-frame 1-10 4 keV energy band; (3) We got all the histograms for every parameter in observer-frame and some parameters in rest-frame; (4) We got all the scatter plots between arbitrary two parameters with at least 5 GRBs; (5) We calculated the linear correlation coefficients and nonlinear correlation ratio between arbitrary two parameters with at least 5 GRBs, and we excluded some correlations with trivial correlations, like T 90 and T 90,i . We do not need to put such correlations in the results. We also considered all the errors using Monte Carlo (MC) method; (6) We did the linear regression between arbitrary two parameters and three parameters with at least 5 GRBs, we also excluded some trivial results. We used MC method to include all the errors. Then we analyzed some interesting results. Because there are many data and results, we made figure sets and several machine readable tables, which can be found in the electric version. Just a small part of the results are shown in this paper as examples. With this complete catalog, we can find more important relations, and we can reveal the GRBs' intrinsic properties.
d The unit is different from the original papers.
e The error is estimated as the central value multiplying 0.1.
f We use the BASTE α peak value -1.1 as common when α is not well constrained.
g We use the BASTE β peak value -2.2 as common when β is not well constrained.
h The value in the original paper is in rest-frame, we change the value using rest-frame divided by (1 + z) to observer-frame.
i We changed the values into logarithm or from logarithm into the normal form.
j The values are calculated using other parameters. Note-All the results are in machine readable table. The description of every parameter is in Section 2.  Note-Correlation coefficients and correlation ratio between two different parameters. The definitions are in Section 4.3. We considered all the errors using MC method (Zou et al. 2018). All the errors are in 1 σ confidence level. All the results are in machine readable table. The description of every parameter is in Section 2. Note-y is dependent variable, x is independent variable, b is the intercept of the linear model, a is the linear coefficient of x. Adjusted R 2 is used to measure the goodness of our regression model, it means the percentage of variance explained considering the parameter freedom. All the errors are in 1 σ error bars. All the results are in machine readable table. The description of every parameter is in Section 2. Note-x1 and x2 are independent variables, a1 and a2 are the linear coefficients of x1 and x2 respectively. y, b and adjusted R 2 are same meanings as in Table 4. All the errors are in 1 σ error bars. All the results are in machine readable table. The description of every parameter is in Section 2. Note-Differences between the current and previous analysis of four relations. Our sample number is obviously bigger than previous analysis.