The mean value of symmetric square $L$-functions

Abstract

We study the first moment of symmetric-square $L$-functions at the critical point in the weight aspect. Asymptotics with the best known error term $O\left(k^{-1∕2}\right)$ were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size $k^{-1∕2}$ in the asymptotic formula and show that the remainder term decays exponentially in $k$. The twisted first moment was evaluated asymptotically by Ng with the error bounded by $l{k}^{-1∕2+ϵ}$. We improve the error bound to $l^{5∕6+ϵ}{k}^{-1∕2+ϵ}$ unconditionally and to $l^{1∕2+ϵ}{k}^{-1∕2}$ under the Lindelöf hypothesis for quadratic Dirichlet $L$-functions.