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This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The one-level density statements and the resulting non-vanishing conclusions are conditional on the Generalized Riemann Hypothesis holding for the L-functions in the families under consideration.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Under GRH, one-level densities for even and odd orthogonal families of automorphic L-functions are established with Fourier support extended to (-3,3), giving the strongest conditional non-vanishing results at the central point.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Conditional on GRH, one-level densities for even and odd orthogonal families of L-functions hold with Fourier support up to 3.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"597d2a640e7d618577d32069fdc2886605b6224b4adff48400277e5dc8158e8d"},"source":{"id":"2605.17012","kind":"arxiv","version":1},"verdict":{"id":"304182ba-954a-4159-9569-2b75c3e72e21","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:54:17.056666Z","strongest_claim":"We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.","one_line_summary":"Under GRH, one-level densities for even and odd orthogonal families of automorphic L-functions are established with Fourier support extended to (-3,3), giving the strongest conditional non-vanishing results at the central point.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The one-level density statements and the resulting non-vanishing conclusions are conditional on the Generalized Riemann Hypothesis holding for the L-functions in the families under consideration.","pith_extraction_headline":"Conditional on GRH, one-level densities for even and odd orthogonal families of L-functions hold with Fourier support up to 3."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17012/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"citation_quote_validity","ran_at":"2026-05-19T19:49:49.418809Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.810169Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:00:41.383224Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T18:51:59.352113Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.189803Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:24.876282Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e2725af5f8057bc848f60acd98889cfec568e97b540fa96882a7d001dbd4ad5d"},"references":{"count":13,"sample":[{"doi":"","year":null,"title":"S. Baluyot, V. Chandee and X. Li,Low-lying zeros of a large orthogonal family of automorphic L-functions, available on arXiv: https://arxiv.org/abs/2310.07606","work_id":"b624e8f9-da2a-4871-b753-efb1ff88e9e2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"O. Barrett, P. Burkhardt, J. DeWitt, R. Dorward, and S.J. Miller,One-level density for holo- morphic cusp forms of arbitrary level.Res. Number Theory, 3: Art. 25, 21,2017","work_id":"997f48e6-da8a-4cda-a818-b3dcf0583095","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"V. Blomer and D. Mili´ cevi´ c.The second moment of twisted modular L-functions. Geom. Funct. Anal., 25(2) (2015), 453 - 516","work_id":"8c58c9fd-c08e-4579-ad00-cb1131e489d3","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"E. Carneiro, V. Chandee, F. Littmann and M. Milinovich,Hilbert spaces and the pair correlation of zeros of the Riemann zeta function, J. Reine Angrew. Math. (2017) 729, 51-79","work_id":"d5d0021a-ef3f-44c2-9c99-35cc5fefb17a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"E. Carneiro, A. Chirre, and M. B. Milinovich,Hilbert spaces and low-lying zeros of L-functions, Adv. Math. 410 (2022), part B, Paper No. 108748, 48 pp","work_id":"3440a6c7-0a53-4874-98bc-af9b76b43443","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"653ff0f77657e21c61080f11df89f4924ddf2fae3e6c51853dde24a422825406","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"}