The first moment of central values of symmetric square $L$-functions in the weight aspect

TL;DR

This paper derives an asymptotic formula for the first moment of symmetric square $L$-functions at the central point in the weight aspect, revealing connections to quadratic fields and secondary main terms involving Dirichlet $L$-values.

Contribution

It provides the first asymptotic formula with power saving for the first moment of symmetric square $L$-functions in the weight aspect, including secondary main terms linked to quadratic fields.

Findings

Established an asymptotic formula with power saving for the first moment.

Extracted secondary main terms involving quadratic field $L$-values.

Revealed dependence of secondary terms on the residue of $k$ modulo 4 and 6.

Abstract

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\frac{1}{2},{\rm{sym}}^2f)$ for $f\in\mathcal{H}_k$ as even $k\rightarrow\infty$, where $\mathcal{H}_k$ is an orthogonal basis of weight-$k$ Hecke eigencuspforms for $SL(2,\mathbb{Z})$. The approach taken in this note allows us to extract two secondary main terms from the error term $O(k^{-\frac{1}{2}})$ in previous studies. More interestingly, our result exhibits a connection between the symmetric square $L$-functions and quadratic fields, which is the main theme of Zagier's work "Modular forms whose coefficients involve zeta-functions of quadratic fields" in 1977. Specifically, the secondary main terms in our asymptotic formula involve central values of Dirichlet $L$-functions of characters $\chi_{-4}$ and $\chi_{-3}$ and depend on the values of $k\,({\rm{mod}}\ 4)$ and $k\,({\rm{mod}}\ 6)$, respectively.