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Zapiski Nauchnykh Seminarov POMI, 2003, Volume 302, Pages 149–167
(Mi znsl927)
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This article is cited in 4 scientific papers (total in 4 papers)
Behavior of automorphic $l$-functions at the points $s=1$ and $s=1/2$
O. M. Fomenko St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
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.
Received: 19.09.2003
Citation:
O. M. Fomenko, “Behavior of automorphic $l$-functions at the points $s=1$ and $s=1/2$”, Analytical theory of numbers and theory of functions. Part 19, Zap. Nauchn. Sem. POMI, 302, POMI, St. Petersburg, 2003, 149–167; J. Math. Sci. (N. Y.), 129:3 (2005), 3898–3909
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https://www.mathnet.ru/eng/znsl927 https://www.mathnet.ru/eng/znsl/v302/p149
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