Riemann hypothesis for period polynomials of modular forms

The period polynomial rf (z) for an even weight k ≥ 4 newform f ∈ Sk (Γ₀(N)) is the generating function for the critical values of L(f, s). It has a functional equation relating rf (z) to r f ( − 1 N z ) . We prove the Riemann hypothesis for these polynomials: that the zeros of rf (z) lie on the circle | z | = 1 / N . We prove that these zeros are equidistributed when either k or N is large.