Abstract
Let \(\phi \) be an even Hecke–Maass cusp form on \({{\text {SL}}}_2({\mathbb {Z}})\) whose L-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of \(\phi \), f and \({\bar{f}}\), where f runs over an orthonormal basis \(H_k\) of Hecke eigen elliptic cusp forms on \({{\text {SL}}}_2({\mathbb {Z}})\) of a fixed weight \(k\geqslant 4\). As an application, we prove a quantitative non-vanishing results on the central values for the family of degree 6 L-functions \(L(s,\phi \times \mathrm{Ad}\,f)\) with f in the union of \(H_k\) \((\mathrm{K}\leqslant k<2\mathrm{K})\) as \(\mathrm{K}\rightarrow \infty \).
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Acknowledgements
The authors would like to thank the anonymous referee for careful reading of the draft. The first author was supported by Grant-in-Aid for Research Activity Start-up 18H05835. The second author was supported by Grant-in-Aid for Scientific research (C) 15K04795.
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Sugiyama, S., Tsuzuki, M. Quantitative non-vanishing of central values of certain L-functions on \({{\text {GL}}}(2)\times {{\text {GL}}}(3)\). Math. Z. 301, 1447–1479 (2022). https://doi.org/10.1007/s00209-021-02886-5
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DOI: https://doi.org/10.1007/s00209-021-02886-5