Average values of symmetric square $L$-functions at the edge of the critical strip

Abstract:

Let ${\mathcal {B}}_{2}^{*}(N)$ be the set of all normalized newforms of weight 2 and level $N$, and let $L({\operatorname {sym}}^{2}f, 1)$ be the symmetric square $L$-function associated to $f\in {\mathcal {B}}_{2}^{*}(N)$. If $N$ is a prime, then there is a positive constant $B$ such that \begin{equation*}\sum _{f\in {\mathcal {B}}_{2}^{*}(N)} L(1,{\operatorname {sym}}^{2}f) = {\frac {\pi ^{4}}{432}} N + O\big (N^{27/28} (\log N)^{B}\big ).\end{equation*} This improves a recent result of Akbary, which requires $45/46$ in place of $27/28$.