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Suppose $P = p^r$, where $p$ is a prime and $r\\equiv 0 (\\textrm{mod} \\ 3)$. Then we have \\[ L\\left( f \\otimes \\chi, \\frac{1}{2}\\right) \\ll_{f, \\epsilon} P^{1/3 +\\epsilon}, \\] where $\\epsilon > 0$ is any positive real number.\n  2.   Suppose $\\chi$ factorizes as $\\chi= \\chi_1 \\chi_2$, where $ \\chi_i$'s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -\\epsilon} \\ll P_i \\ll P^{1/2 +"},"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":"1706.03985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-06-13T10:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"d745108e8ac5ce39e2280ae2e6f0fb9b580bf733f77f393db1c4c691c9024a7a","abstract_canon_sha256":"c5770b91761c144e47dfb41d70db8c9c81ab46d5863787cd0969d06eec8acd3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:27.822429Z","signature_b64":"ZRWE8OB0XmamF45amh42K6OAyfkVA86W4KokWhD/VZ73WCQAk7HCdrk8LR5Bw0/PLNXnFYuhWvK/6Ox1uoSnBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b3678a9cfccc70b5f759216962c86a9a37883779952b3543ac5d1f67ec6f24a","last_reissued_at":"2026-05-18T00:42:27.821829Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:27.821829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"W\\lowercase{eyl} \\lowercase {bound for $p$-power twist of} $GL(2)$ L-\\lowercase{functions }","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ritabrata Munshi, Saurabh Kumar Singh","submitted_at":"2017-06-13T10:13:31Z","abstract_excerpt":"Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \\mathbb{Z})$ . Let $\\chi$ be a primitive character of modulus $P$. We shall prove the following results:\n 1. Suppose $P = p^r$, where $p$ is a prime and $r\\equiv 0 (\\textrm{mod} \\ 3)$. Then we have \\[ L\\left( f \\otimes \\chi, \\frac{1}{2}\\right) \\ll_{f, \\epsilon} P^{1/3 +\\epsilon}, \\] where $\\epsilon > 0$ is any positive real number.\n  2.   Suppose $\\chi$ factorizes as $\\chi= \\chi_1 \\chi_2$, where $ \\chi_i$'s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -\\epsilon} \\ll P_i \\ll P^{1/2 +"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03985","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":""},"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":"1706.03985","created_at":"2026-05-18T00:42:27.821908+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.03985v1","created_at":"2026-05-18T00:42:27.821908+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03985","created_at":"2026-05-18T00:42:27.821908+00:00"},{"alias_kind":"pith_short_12","alias_value":"RM3HRKOPZTDQ","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"RM3HRKOPZTDQWX3V","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"RM3HRKOP","created_at":"2026-05-18T12:31:39.905425+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/RM3HRKOPZTDQWX3VSILJMLEGVG","json":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG.json","graph_json":"https://pith.science/api/pith-number/RM3HRKOPZTDQWX3VSILJMLEGVG/graph.json","events_json":"https://pith.science/api/pith-number/RM3HRKOPZTDQWX3VSILJMLEGVG/events.json","paper":"https://pith.science/paper/RM3HRKOP"},"agent_actions":{"view_html":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG","download_json":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG.json","view_paper":"https://pith.science/paper/RM3HRKOP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.03985&json=true","fetch_graph":"https://pith.science/api/pith-number/RM3HRKOPZTDQWX3VSILJMLEGVG/graph.json","fetch_events":"https://pith.science/api/pith-number/RM3HRKOPZTDQWX3VSILJMLEGVG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG/action/storage_attestation","attest_author":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG/action/author_attestation","sign_citation":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG/action/citation_signature","submit_replication":"https://pith.science/pith/RM3HRKOPZTDQWX3VSILJMLEGVG/action/replication_record"}},"created_at":"2026-05-18T00:42:27.821908+00:00","updated_at":"2026-05-18T00:42:27.821908+00:00"}