Central value of the symmetric square L-functions related to Hecke–Maass forms

Abstract

Let {ϕj(z) : j ≥ 1} be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4 + tj2. For each ϕj(z), we have the automorphic L-function L(s, sym2ϕj), which is called the symmetric square L-function associated to ϕj. In this paper, we consider the average estimate of L(1/2, sym2ϕj) and prove that, for sufficiently large T, the estimate \( {\displaystyle {\sum}_{T<{t}_j\le T+M}{\left|L\left(1/2,{\mathrm{sym}}^2{\phi}_j\right)\right|}^2\ll {T}^{1+\varepsilon }M} \) holds for T1/3 + e ≤ M ≤ T1 − e.