The Ramanujan Journal

, Volume 46, Issue 3, pp 775–794 | Cite as

The first moment of central values of symmetric square L-functions in the weight aspect

  • Shenhui LiuEmail author


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.


Holomorphic Hecke eigenforms Symmetric square L-functions Weight aspect Quadratic fields 

Mathematics Subject Classification

11F11 11F66 11F67 



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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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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