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Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\\times g) $ where $f\\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a result, we recover the bound $ L(1/2,f\\times g) \\ll_\\varepsilon (N + \\sqrt{M}) N^\\varepsilon M^\\varepsilon $ when $g$ is dihedral. The first moment method also applies to the special derivative $L'(1/2,f\\times g)$ under the assumption that it is non-negative for all $f\\in S^*_k(N)$."},"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":"1207.3421","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-14T13:04:28Z","cross_cats_sorted":[],"title_canon_sha256":"637cd044683a7ae4a2894f89e51c61b61e750fa44f0752b15954b62616fb0d18","abstract_canon_sha256":"da1e8bf78153654e6564c870e8348783f897bd5213973a6fcb40df32f85eaa82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:07.022844Z","signature_b64":"H4uKIZsoVQyRskDG6t6WzC7vV52peYVLhbhx2wvO5Ed2rQqGW3C9igHL7pl/zjjovyUsTUjENHe+0sSv/5QEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cf1e1a905f079165a20df8b8a9e613866f5c5530fea458538e03f99105f9bf7","last_reissued_at":"2026-05-18T03:43:07.022261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:07.022261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"First moment of Rankin-Selberg central L-values and subconvexity in the level aspect","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nicolas Templier, Roman Holowinsky","submitted_at":"2012-07-14T13:04:28Z","abstract_excerpt":"Let $1\\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\\times g) $ where $f\\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a result, we recover the bound $ L(1/2,f\\times g) \\ll_\\varepsilon (N + \\sqrt{M}) N^\\varepsilon M^\\varepsilon $ when $g$ is dihedral. The first moment method also applies to the special derivative $L'(1/2,f\\times g)$ under the assumption that it is non-negative for all $f\\in S^*_k(N)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3421","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":""},"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":"1207.3421","created_at":"2026-05-18T03:43:07.022392+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.3421v3","created_at":"2026-05-18T03:43:07.022392+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3421","created_at":"2026-05-18T03:43:07.022392+00:00"},{"alias_kind":"pith_short_12","alias_value":"JTY6DKIF6B4R","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"JTY6DKIF6B4RMWRA","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"JTY6DKIF","created_at":"2026-05-18T12:27:11.947152+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/JTY6DKIF6B4RMWRA36FYVHTBHB","json":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB.json","graph_json":"https://pith.science/api/pith-number/JTY6DKIF6B4RMWRA36FYVHTBHB/graph.json","events_json":"https://pith.science/api/pith-number/JTY6DKIF6B4RMWRA36FYVHTBHB/events.json","paper":"https://pith.science/paper/JTY6DKIF"},"agent_actions":{"view_html":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB","download_json":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB.json","view_paper":"https://pith.science/paper/JTY6DKIF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.3421&json=true","fetch_graph":"https://pith.science/api/pith-number/JTY6DKIF6B4RMWRA36FYVHTBHB/graph.json","fetch_events":"https://pith.science/api/pith-number/JTY6DKIF6B4RMWRA36FYVHTBHB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB/action/storage_attestation","attest_author":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB/action/author_attestation","sign_citation":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB/action/citation_signature","submit_replication":"https://pith.science/pith/JTY6DKIF6B4RMWRA36FYVHTBHB/action/replication_record"}},"created_at":"2026-05-18T03:43:07.022392+00:00","updated_at":"2026-05-18T03:43:07.022392+00:00"}