On the behavior of automorphic $L$-functions at the center of the critical strip

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 300-311

Résumé

Let $\mathscr F$ be the Hecke eigenbasis of the space $S_2(\Gamma_0(p))$ of $\Gamma_0(p)$-cusp forms of weight 2. Let $p$ be a prime. Let $\mathscr H_f(s)$ be the Hecke $L$-series of form $f\in\mathscr F$. The following statements are proved: $$ \sum_{f\in\mathscr F}\mathscr H_f\left(\frac12\right)=\zeta(2)\frac p{12}+O\left(p^{\frac{31}{32}+\varepsilon}\right) $$ and $$ \sum_{f\in F}\mathscr H_f\left(\frac12\right)^2=\frac{\zeta(2)^3}{\zeta(4)}\frac p{12}\log p+O(p\log\log p). $$ We also give a correct proof of a previous author's theorem on automorphic $L$-functions.