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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On the distribution of values of $ L(1,\mathrm{sym}^2f)$

Author: O. M. Fomenko
Translated by: G. V. Kuz'mina and O. M. Fomenko
Original publication: Algebra i Analiz, tom 17 (2005), nomer 6.
Journal: St. Petersburg Math. J. 17 (2006), 1031-1046
MSC (2000): Primary 11M41
Published electronically: September 20, 2006
MathSciNet review: 2202450
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Abstract: Let $ S_k(\mathrm{SL}(2,\mathbb{Z}))^+$ be the set of holomorphic Hecke eigencuspforms $ f$ of weight $ k$ with respect to $ \mathrm{SL}(2,\mathbb{Z})$. Let $ L(s,\mathrm{sym}^2f)$ be the symmetric square of the Hecke L-function of a cusp form $ f$. The moments of $ L(1,\mathrm{sym}^2f)$, $ f\in S_k(\mathrm{SL}(2,\mathbb{Z}))^+$, are computed for a pure imaginary order. The limiting distribution of $ \log L(1,\mathrm{sym}^2 f)$, $ f\in S_k(\mathrm{SL}(2,\mathbb{Z}))^+$, is studied in the weight aspect. Namely, the limiting distribution function, the limiting characteristic function and its Euler product are investigated, and the rate of convergence of frequencies to the limiting distribution is measured.

As a consequence, new facts on the limiting distribution of $ \mathrm{SL}(1,\mathrm{sym}^2f)$ are obtained not only in the case of the holomorphic Hecke eigencuspforms $ f$, but also in the case of the Hecke-Maass eigencuspforms $ f$.

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Additional Information

O. M. Fomenko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Automorphic $L$-function, symmetric square $L$-function, large sieve, distribution function, characteristic function
Received by editor(s): March 10, 2005
Published electronically: September 20, 2006
Additional Notes: Partially supported by RFBR (grant no. 05-01-00930).
Dedicated: Dedicated to Yuriĭ Vladimirovich Linnik’s 90th birthday anniversary
Article copyright: © Copyright 2006 American Mathematical Society

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