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

Abstract

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\mathrm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\frac{1}{2},{\mathrm{sym}}^{2}f)$ for $f\in {\mathcal{H}}_k$ as even $k\to \infty$, 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 $k(\mathrm{mod}4)$ and $k(\mathrm{mod}6)$, respectively.

17 pages; typos corrected; minor changes of writing in text