{"paper":{"title":"Critical Zeros and Unconditional Mean Value Theorems for twisted $\\hbox{PGL}(2)$ and $\\hbox{PGL}(3)$ $\\mathrm{L}$-functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Brian Conrey, Caroline L. Turnage-Butterbaugh, Chung-Hang Kwan, Yongxiao Lin","submitted_at":"2026-07-01T00:13:25Z","abstract_excerpt":"Let $\\Pi_{0}$ be a cuspidal automorphic representation of $\\mathrm{PGL}_{3}(\\mathbb{A}_{\\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\\to \\infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, \\Pi_{0}\\,\\times\\, \\chi)$ lie on the critical line, where $\\chi$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $\\Pi_{0}$ is self-dual, and otherwise holds under a mild condition.\n  The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00282","kind":"arxiv","version":1},"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/2607.00282/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"}