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Following [Zac20], we estimate the first moment of $L(\\frac{1}{2}, \\pi \\otimes \\pi_1 \\otimes \\pi_2)$ twisted by the Hecke eigenvalues $\\lambda_{\\pi}(\\mathfrak{l})$, where $\\pi$ runs over unitary automorphic representations of finite conductor div"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2501.10418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-12-27T18:56:07Z","cross_cats_sorted":[],"title_canon_sha256":"871ba9256900f9eaddf9fd62d417a40f66900fce0851724741cd90f7e88f8c7c","abstract_canon_sha256":"9e45f48da66194a77b71394bcdead7bbe5645710e32fc47320cf909955d1b517"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:02:27.636752Z","signature_b64":"qXwspXq2U9g120LYgfYfCKIuu+RDXlLtHVLxTxeba8X7LGRbjkoKJPsVI/BBsDGJ/OcumMhcXriUc7GbIb90Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f2ba1de9320e9ab82a43d3cc680e7f9ead73a3cae1f20d58b7bac7deae78346","last_reissued_at":"2026-07-05T10:02:27.636356Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:02:27.636356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral Reciprocity for the first moment of triple product $L$-functions and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xinchen Miao","submitted_at":"2024-12-27T18:56:07Z","abstract_excerpt":"Let $F$ be a number field with adele ring $\\mathbb{A}_F$, $\\pi_1, \\pi_2$ be two fixed unitary automorphic representations of $\\mathrm{PGL}_2(\\mathbb{A}_F)$ with finite coprime analytic conductor $\\mathfrak{u}$ and $\\mathfrak{v}$, $\\mathfrak{q},\\mathfrak{l}$ be two coprime integral ideals with $(\\mathfrak{q} \\mathfrak{l}, \\mathfrak{u} \\mathfrak{v})=1$. Following [Zac20], we estimate the first moment of $L(\\frac{1}{2}, \\pi \\otimes \\pi_1 \\otimes \\pi_2)$ twisted by the Hecke eigenvalues $\\lambda_{\\pi}(\\mathfrak{l})$, where $\\pi$ runs over unitary automorphic representations of finite conductor div"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.10418","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/2501.10418/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2501.10418","created_at":"2026-07-05T10:02:27.636415+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.10418v1","created_at":"2026-07-05T10:02:27.636415+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.10418","created_at":"2026-07-05T10:02:27.636415+00:00"},{"alias_kind":"pith_short_12","alias_value":"F4V2DXUTEDU2","created_at":"2026-07-05T10:02:27.636415+00:00"},{"alias_kind":"pith_short_16","alias_value":"F4V2DXUTEDU2XAVE","created_at":"2026-07-05T10:02:27.636415+00:00"},{"alias_kind":"pith_short_8","alias_value":"F4V2DXUT","created_at":"2026-07-05T10:02:27.636415+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H","json":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H.json","graph_json":"https://pith.science/api/pith-number/F4V2DXUTEDU2XAVEHU6MNAHH7H/graph.json","events_json":"https://pith.science/api/pith-number/F4V2DXUTEDU2XAVEHU6MNAHH7H/events.json","paper":"https://pith.science/paper/F4V2DXUT"},"agent_actions":{"view_html":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H","download_json":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H.json","view_paper":"https://pith.science/paper/F4V2DXUT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.10418&json=true","fetch_graph":"https://pith.science/api/pith-number/F4V2DXUTEDU2XAVEHU6MNAHH7H/graph.json","fetch_events":"https://pith.science/api/pith-number/F4V2DXUTEDU2XAVEHU6MNAHH7H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H/action/storage_attestation","attest_author":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H/action/author_attestation","sign_citation":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H/action/citation_signature","submit_replication":"https://pith.science/pith/F4V2DXUTEDU2XAVEHU6MNAHH7H/action/replication_record"}},"created_at":"2026-07-05T10:02:27.636415+00:00","updated_at":"2026-07-05T10:02:27.636415+00:00"}