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If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the congruence subgroups $\\Gamma_0(p)$ and $\\Gamma_0(q)$ (resp) with trivial nebentypus, then for all $\\epsilon >0$ we show that there exists an $A >0$ such that $$ L\\left(\\frac{1}{2}+it, f \\times g \\right) \\ll_{\\epsilon,\\mu_f, \\mu_g}(1+|t|)^A \\frac{(pq)^{1/2+\\epsilon}}{\\max\\{p,q \\}^{\\frac{1}{64}}}. $$ The dependence on $\\mu_f$ and $\\mu_g$, the parameters at infini"},"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":"1807.11092","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-07-29T18:08:08Z","cross_cats_sorted":[],"title_canon_sha256":"bcc0735a866d38a1de6226c3626ae7dfb89213caad46e816ba87895c856b6881","abstract_canon_sha256":"c087dc8296049e24f30ef0269ac89ab5f686c2e64aa1468b96e0531be65147fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:32.971972Z","signature_b64":"lt8jXaGlkyQCBgWqy+V9hBlt6R4S3DmW0kc6Cmv85eSKGeL/USnsc2GPLE7Eq2GqHTqcWQKwplYVxvagclx4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38d7bc4909e4b76e45a03cadd6307a99b65b4c06f2ba9196d9e95c9e32a7e534","last_reissued_at":"2026-05-18T00:09:32.971570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:32.971570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sub-convexity problem for Rankin-Selberg $L$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chandrasekhar Raju","submitted_at":"2018-07-29T18:08:08Z","abstract_excerpt":"We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. 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