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Behavior of automorphic \(L\)-functions at the center of the critical strip. (English. Russian original) Zbl 1130.11318

J. Math. Sci., New York 118, No. 1, 4910-4917 (2003); translation from Zap. Nauchn. Semin. POMI 276, 300-311 (2001).
Summary: 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 Theorem 2 in the author’s paper [J. Math. Sci., New York 110, No. 6, 3143–3149 (2002); translation from Zap. Nauchn. Semin. POMI 263, 193–204 (2000; Zbl 1037.11031)] on automorphic \(L\)-functions.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F11 Holomorphic modular forms of integral weight
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Citations:

Zbl 1037.11031
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