, Volume 118, Issue 1, pp 4910-4917

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

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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.