Abstract
We establish an asymptotic formula with arbitrary power saving for the first moment of the symmetric square L-functions \(L(s,\mathrm{sym}^2f)\) at \(s=\frac{1}{2}\) for \(f\in \mathcal {H}_k\) as even \(k\rightarrow \infty \), where \(\mathcal {H}_k\) is an orthogonal basis of weight-k Hecke eigen cusp forms for \(SL(2,\mathbb {Z})\). The approach taken allows us to extract two secondary main terms from the best-known error term \(O(k^{-\frac{1}{2}})\). Moreover, our result exhibits a connection between the symmetric square L-functions and quadratic fields, which is the main theme of Zagier’s work Modular forms whose coefficients involve zeta-functions of quadratic fields in 1977.
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Notes
In the last line of Eq. (2) of [8], \(2^{k-2}\) should be \(2^{k-1}\).
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Acknowledgements
The author thanks Professor Wenzhi Luo for stimulating conversations and helpful comments. The author thanks Professors Dorian Goldfeld, Roman Holowinsky, and Kannan Soundararajan for their interest in this work. Thanks also go to the referee for the thorough reading and helpful suggestions.
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Liu, S. The first moment of central values of symmetric square L-functions in the weight aspect. Ramanujan J 46, 775–794 (2018). https://doi.org/10.1007/s11139-017-9921-6
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DOI: https://doi.org/10.1007/s11139-017-9921-6