Zeros of period polynomials for symmetric power <i>L</i> -functions

Suppose that k and N are positive integers. Let f be a newform on [Formula: see text] of weight k with L-function [Formula: see text]. Previous works have studied the zeros of the period polynomial [Formula: see text], which is a generating function for the critical values of [Formula: see text] and has a functional equation relating z and [Formula: see text]. In particular, [Formula: see text] satisfies a version of the Riemann hypothesis: all of its zeros are on the circle of symmetry [Formula: see text]. In this paper, for a positive integer m, we define a natural analog of [Formula: see text] for the mth symmetric power L-function of f when N is squarefree. Our analog also has a functional equation relating z and [Formula: see text]. We prove the corresponding version of the Riemann hypothesis when k is large enough. Moreover, when [Formula: see text], we prove our result when N is large enough.