{"paper":{"title":"Expected Values of $L$-functions Away from the Central Point","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chantal David, Patrick Meisner","submitted_at":"2023-05-24T13:11:26Z","abstract_excerpt":"We compute the expected value of Dirichlet $L$-functions defined over $\\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \\in (0,1)$. We find a transition term at the point $s=\\frac{1}{3}$, reminiscent of the transition at the point $s=\\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindel\\\"of hypothesis. We show that at $s=\\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.15120","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.15120/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"}