{"paper":{"title":"The first moment of central values of symmetric square $L$-functions in the weight aspect","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shenhui Liu","submitted_at":"2016-10-24T21:05:08Z","abstract_excerpt":"In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\\frac{1}{2},{\\rm{sym}}^2f)$ for $f\\in\\mathcal{H}_k$ as even $k\\rightarrow\\infty$, where $\\mathcal{H}_k$ is an orthogonal basis of weight-$k$ Hecke eigencuspforms for $SL(2,\\mathbb{Z})$. The approach taken in this note allows us to extract two secondary main terms from the error term $O(k^{-\\frac{1}{2}})$ in previous studies. More interestingly, our resul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07652","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}