{"paper":{"title":"On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi Qi","submitted_at":"2025-06-10T08:12:05Z","abstract_excerpt":"Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that there is a constant $ \\gamma (Q) $ expressed as a certain Euler product associated to $Q$ such that at least $ \\gamma (Q) / 11 $ of the Rankin--Selberg special $L$-values $L (1/2+it_j, Q \\otimes u_j)$ for $ t_j \\leqslant T$ do not vanish as $T \\rightarrow \\infty$. Further, we show that the non-vanishing proportion is at least $\\gamma (Q) \\cdot (4\\mu-3) / (4\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08546","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.08546/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}