The “Riemann Hypothesis” is true for period polynomials of almost all newforms

*Correspondence: [email protected] 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Full list of author information is available at the end of the article Abstract The period polynomial rf (z) for a weight k ≥ 3 and level N newform f ∈ Sk (Γ0(N),χ ) is the generating function for special values of L(s, f ). The functional equation for L(s, f ) induces a functional equation on rf (z). Jin, Ma, Ono, and Soundararajan proved that for all newforms f of even weight k ≥ 4 and trivial nebentypus, the “Riemann Hypothesis” holds for rf (z): that is, all roots of rf (z) lie on the circle of symmetry |z| = 1/ √ N. We generalize their methods to prove that this phenomenon holds for all but possibly finitely many newforms f of weight k ≥ 3 with any nebentypus. We also show that the roots of rf (z) are equidistributed if N or k is sufficiently large.