Behavior of automorphic $l$-functions at the points $s=1$ and $s=1/2$

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 149-167

Résumé

Let $S_k(N)^+$ be the set of primitive cusp forms of even weight $k$ for $\Gamma_0(N)$ and let $L(s,\operatorname{sym}^2f)$ be the symmetric square $L$-function $L(s,f)$ of a form $f\in S_k(N)^+$. The moments of the variable $L(s,\operatorname{sym}^2f)$, $f\in S_2(N)^+$, are computed for $N=p$, and the corresponding limiting distribution is determined in $N$-aspect. Let $f\in S_k(1)^+$, $g\in S_l(1)^+$, and $\omega_f=\Gamma(k-1)/(4\pi)^{k-1}{\langle f,f\rangle}$. Asymptotic formulas for $\sum_{f\in S_k(1)^+}\omega_f L\Bigl(\frac12,\operatorname{sym}^2 f\Bigr)$ and $\sum_{f\in S_k(1)^+}\omega_f L\Bigl(\frac12,f\otimes g\Bigr)$ as $k\in\infty$ are obtained.