Behavior of Automorphic L-Functions at the Center of the Critical Strip

Abstract

Let $\mathcal{F}$ 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 $\mathcal{H}_f (s)$ be the Hecke L-series of form $f \in \mathcal{F}$. The following statements are proved:

$$\sum\limits_{f \in \mathcal{F}} {\mathcal{H}_f \left( {\frac{1}{2}} \right)} = \zeta (2)\frac{p}{{12}} + O\left( {p^{\frac{{31}}{{32}} + \varepsilon } } \right)$$

and

$$\sum\limits_{f \in \mathcal{F}} {\mathcal{H}_f \left( {\frac{1}{2}} \right)} ^2 = \frac{{\zeta (2)^3 }}{{\zeta (4)}}\frac{p}{{12}}\log p + O\left( {p\log \log p} \right).$$

We also give a correct proof of a previous author's theorem on automorphic L-functions. Bibliography: 12 titles.