On the central value of symmetric square L-functions

Abstract

Let S k (N, χ) be the space of cusp forms of weight k, level N and character χ. For $$f \in S_k(N, \chi)$$ let L(s, sym2 f) be the symmetric square L-function and $$L(s, f \otimes f)$$ be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym2 f) and $$L(s, f \otimes f)$$ over a basis of S k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym2 f) does not vanish.