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arXiv · 2609.16893

Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions in the level aspect

Zhiwei Liu
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Abstract

In this paper, we establish an asymptotic formula of mixed moments of $L(1/2,f)$ and $L(1/2,\mathrm{sym}^2 f)$ for cusp forms $f$ of weight $k$, prime level $q$ and real primitive nebentypus $χ$ modulo $q$ in the level aspect with a power-saving error term. As an application, we deduce the simultaneous non-vanishing of $L(1/2, f)$ and $L(1/2,\mathrm{sym}^2 f)$ for sufficiently large prime levels.

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Zhiwei Liu. 2026-09-15. Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions in the level aspect. https://doi.org/10.1090/proc%2F17898

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