The"Riemann Hypothesis"is True for Period Polynomials of Almost All Newforms

The period polynomial $r_f(z)$ for a weight $k \geq 3$ newform $f \in S_k(\Gamma_0(N),\chi)$ is the generating function for special values of $L(s,f)$. The functional equation for $L(s, f)$ induces a functional equation on $r_f(z)$. Jin, Ma, Ono, and Soundararajan proved that for all newforms $f$ of even weight $k \ge 4$ and trivial nebetypus, the"Riemann Hypothesis"holds for $r_f(z)$: that is, all roots of $r_f(z)$ lie on the circle of symmetry $|z| =1/\sqrt{N}$. We generalize their methods to prove that this phenomenon holds for all but possibly finitely many newforms $f$ of weight $k \ge 3$ with any nebentypus. We also show that the roots of $r_f(z)$ are equidistributed if $N$ or $k$ is sufficiently large.


Introduction and Statement of Results
Let f ∈ S k (Γ 0 (N), χ) be a newform of weight k, level N, and nebentypus χ. Associated to f is an L-function L(s, f ), which can be normalized so that the completed L-function for some ǫ(f ) ∈ C with |ǫ(f )| = 1. The period polynomial associated to f is the degree k − 2 polynomial defined by (1.3) r f (z) · · = i∞ 0 f (y)(y − z) k−2 dy.
Date: March 14, 2018. 1 When k ≥ 3, the period polynomial r f (z) is nonconstant, so one can consider where the roots of r f (z) are located. To this end, we use the functional equation (1.2) to observe that Thus, if ρ is a root of r f (z), then 1 Nρ is also a root. Much like the behavior of the nontrivial zeroes of L(s, f ) predicted by the Generalized Riemann Hypothesis, one can consider whether all the roots of r f (z) lie on the curve of symmetry of the roots: in this case, the circle |ρ| = 1/ √ N . It is natural to expect the following conjecture, which is supported by extensive numerical evidence.
El-Guindy and Raji [3] proved this for Hecke eigenforms on SL 2 (Z) with full level (N = 1, for which the circle of symmetry is |z| = 1). They were inspired by the work of Conrey, Farmer, andİmamoglu [2], who showed an analogous result for the odd parts of these period polynomials, again with full level.
Recent work by Jin, Ma, Ono, and Soundararajan [5] proved the conjecture for all newforms of even weight k ≥ 4 and trivial nebentypus. They also showed that the roots of r f (z) are equidistributed on the circle of symmetry for sufficiently large N or k. Using similar methods, Löbrich, Ma, and Thorner [8] proved an analogous result for polynomials generating special values of L(s, M) for a sufficiently well-behaved class of motives M with odd weight and even rank.
In this paper, we generalize the methods of [5] to prove the conjecture for all but possibly finitely many newforms.
Theorem 1.1. The "Riemann Hypothesis" for period polynomials holds for all but possibly finitely many newforms with weight k ≥ 3 and nontrivial nebentypus.
Remark. Note that for k < 3, the period polynomial is a constant. Therefore, Theorem 1.1 is essentially the best result for which one could hope, since an effective computation can check that Theorem 1.1 also holds for the finitely many possible exceptions. We denote the set of these finitely many newforms as S, which consists of the following: (1) For k = 5, all newforms with level N ≤ 10331.
(2) For k ≥ 6, all newforms with level N ≤ C(k), where C(k) is a constant given by tables at the end of Section 4 and 5. We know of no counterexamples to Theorem 1.1.
We also show that the roots of r f (z) are equidistributed on the circle of symmetry for sufficiently large N or k. Theorem 1.2. Let f ∈ S k (Γ 0 (N), χ) be a newform of weight k ≥ 4, level N, and nebentypus χ such that f / ∈ S. Then, the following are true: (i) Suppose that k = 4, and let z 1 , z 2 denote the roots of r f (z). Then for any real ε > 0, where the implied constant depends only on ε and is effectively computable.
(ii) Suppose that k = 5. There exists c f ∈ R such that the arguments of the roots of r f (z) can be written as where θ ℓ denotes the unique solution mod 2π of and the implied constant depends only on ε and is effectively computable. (iii) Suppose that k > 5. There exists c f ∈ R such that the arguments of the roots of r f (z) can be written as Here, θ ℓ is the unique solution mod 2π to the equation and the implied constant is absolute and effectively computable.
In Section 2, we introduce notation and lemmas that we will be using in our proof. In Section 3, we will prove our main results for k = 3, 4, and 5 using ad hoc arguments. For larger k, we prove Theorem 1.1 in Section 4 (the case of k even) and Section 5 (the case of k odd), and we prove Theorem 1.2 in Section 6. Finally, in Section 7, we detail our Sage computations suggesting that the roots of the period polynomial of the newform are all on the circle |z| = 1/ √ 11. This newform f is in our finite set S of possible exceptions, which suggests that Theorem 1.1 should be true even for newforms in S.

Preliminaries
Throughout this section, we assume that f ∈ S k (Γ 0 (N), χ) is a newform of weight k ≥ 3, level N, and arbitrary nebentypus χ. We note that the nebentypus character will be essentially invisible throughout our proof, other than the fact that it determines the level of f . We now define some notation related to r f (z) and prove lemmas about the values of Λ(s, f ) and L(s, f ) along the real line. The lemmas will be very similar in spirit to those proven in [5].
Define δ to satisfy where z 1 2 denotes r 1 2 e θi/2 for z = re θi and 0 ≤ θ < 2π. Using (1.2), one can compute Therefore, if t f (z) = 0, then t f 1 z = 0. Additionally, for |z| = 1, we also have t f (z) = t f (z), so t f (z) is real for |z| = 1. Note that t f (z) = 0 if and only if r f ( z i √ N ) = 0. Therefore, to prove Theorems 1.1 and 1.2, it suffices to show that all roots of t f (z) lie on the circle |z| = 1 and are equidistributed.
We will require the following monotonicity result.
Also, for all 0 < a < b, Proof. As Λ(s, f ) is entire of order 1, we apply the Hadamard factorization theorem to write where the sum is taken over all roots ρ of Λ(s, f ). By [4, Proposition 5.7 (3)], we have that from which the lemma follows. We also prove a useful inequality on ratios of L-function values.
Proof. We have that n s , then Deligne's bound on the eigenvalues of the Hecke operators on S k (Γ 0 (N), χ) states that where Λ(n) denotes the von Mangoldt function; for a reference, see [9,Theorem 2.32]. Therefore, we have that Now the lemma follows from the inequality |e x − 1| ≤ e |x| − 1.
Finally, we show a lemma that serve as our main means of proving Theorem 1.1 for period polynomials. Lemma 2.3. Let sgn(r) equal −1 for negative real numbers r, 1 for positive real numbers r, and 0 for r = 0. If there exist real numbers 0 ≤ θ 1 < θ 2 < · · · < θ k < 2π such that either then all solutions to t f (z) = 0 satisfy |z| = 1.
Proof. First, t f (z) is real for |z| = 1, so sgn(t f (e iθ j )) is well defined. Now, by the Intermediate Value Theorem, there exist θ ∈ (θ j , θ j+1 ) such that t f (e iθ ) = 0 for all 1 ≤ j ≤ k − 1. This gives us k − 1 roots of t f (z) that lie on |z| = 1. When k is even, we also get a root in the range (θ k , θ 1 + 2π) by the Intermediate Value Theorem. When k is odd, we may redefine the square root in order to move the discontinuity into an interval outside of [θ k , θ 1 + 2π]. This would only affect the sign of t f (z). By the intermediate value theorem, this shows the existence of a zero with argument in the range [θ k , θ 1 + 2π] as desired. As t f (z) = 0 for at most k values of z, the above argument shows that we have found all of them.

Proof for Remaining Even Weights
In this section, we will show Theorem 1.1 for all even weights k ≥ 6. Throughout the section, we will restrict our attention to those z such that |z| = 1. For simplicity, let m = k−2 2 , and define This satisfies t f (z) = P f (z) + P f (z) = 2 Re(P f (z)).
Next, define Note that sgn(Q f (z)) = sgn(P f (z)). As in [5], rewrite where we define For z = e iθ , note that where C is a fixed constant depending on δ and L(2m+ 1, f ). Therefore, we can pick k values of z on the circle |z| = 1 such that the previous expression has argument ℓπ for integers ℓ.
The value of Q f (z) at these points have alternating positive and negative real part with magnitude at least |L(2m + 1, f )| exp − 2π √ N . By Lemma 2.3, it suffices to show that To bound S 1 (z), we use Lemma 2.2 in the form L(2m+1−n,f ) For the term n = m − 1 in the above expression, we use the bound ζ(3/2) 2 − 1 ≤ 35/6. For 0 ≤ n ≤ m − 2, note that 2 x (ζ(1/2 + x) 2 − 1) is decreasing for x ≥ 2. Therefore, for 0 ≤ n ≤ m − 2, we find that Now, we combine the above estimates with S 2 (z) to obtain To finish, we estimate |S 3 (z)| using Lemma 2.1 and then 2.2.
By using (4.3) and (4.4), it suffices to verify (4.5) For each value of m in the first row on the following table, the value N(m) is such that inequality (4.5) holds for all N ≥ N(m). Note that the case m = 1 was done in Section 3. Therefore, for all m ≥ 29, N(m) = 1. This completes our proof of Theorem 1.1 for k even.

Proof for Remaining Odd Weights
In this section, we will show Theorem 1.1 for all odd weights k ≥ 7. As in the above section, we will restrict our attention to those z such that |z| = 1. For simplicity, let m = k−3 2 , and define P f (z) = m n=0 k − 2 n z m−n+ 1 2 δΛ(n + 1, f ) so t f (z) = 2 Re(P f (z)) As in the above section, define where S 1 (z), S 2 (z), and S 3 (z) are defined as follows.
As in Section 4, it suffices to show that The proof of this will proceed in a very similar way to that of the above section. Note that the function 2 x (ζ(1 + x) 2 − 1) is decreasing for x ≥ 1. By Lemma 2.2, for 0 ≤ n ≤ m − 1, we can bound By Lemma 2.2, we have Now we use Lemma 2.1 to bound |L(m + 2, f )|.
Therefore, we have that For each value of m in the first row on the following table, the value N(m) is such that inequality (5.4) holds for all N ≥ N(m). Note that the cases m = 0, 1 was done in Section 3.

Equidistribution of Roots for Large Weights
Let k ≥ 6. For even k, set m = k−2 2 . Then, the arguments in the previous sections show that for z = e iθ , then Consider the θ ℓ such that mθ ℓ + C − 2π √ N sin θ ℓ = π 2 + ℓπ. Then it is simple to verify that for some constant D, the two values θ ℓ ± D 2 m √ N , Re(Q f (z)) has different signs. This completes the proof for even k.
All the coefficients of f are real, and we have ǫ(f ) = 1. In light of the functional equation L(s, f ) = L(k − 1 − s, f ), we can use Sage to compute the critical values of L(s, f ) and thereby obtain r f (z). We calculate that the roots of r f (z) are z 1 ≈ −0.294570496142963 − 0.0643219535709181i, z 2 ≈ −0.204098252273756 + 0.221930156418385i, z 3 ≈ 0.301511344577764i, z 4 ≈ 0.204098252273756 + 0.221930156418385i, and z 5 ≈ 0.294570496142963 − 0.0643219535709181i. All five roots have absolute value ≈ 0.301511344577764 ≈ 1/ √ 11, as expected given the statement of Theorem 1.1.