Spectral decomposition formula and moments of symmetric square L-functions

Abstract

We prove a spectral decomposition formula for averages of Zagier L-series in terms of moments of symmetric square L-functions associated to Maass and holomorphic cusp forms of levels 4, 16, 64.

An explicit formula for the second moment of Maass form symmetric square $L$-functions

“…The Fourier-Whittaker expansion for the given Eisenstein series is obtained in [2,Theorem 2.3]. Here we just state the final result.…”

“…Changing (u) from (u) > 1/2 to (u) < 1/2, these poles "jump" through the line of integration, resulting in additional residues (see [2,Lemma 6.4]). Computing the residues at the points (5.51) and using the fact that F (t) is an even function, we prove (5.47).…”

“…It turns out that this double sum can be written in terms of Gauss sums, and consequently can be evaluated explicitly (see [2,Section 3]). As a result, we have an expression for K(n, l; q) in terms of generalized Kloosterman sums; see Subsection 2.2.…”

“…Consequently, we need to compute Fourier coefficients of Eisenstein series for all singular cusps. This was done in [2] and here we just state the final formulas. Lemma 2.9 ([2, Lemma 5.1]).…”