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Let $\\rho$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $\\pi$ a cuspidal automorphic representation of GL$_2(\\mathbb A_F)$, such that the $L$-functions $L(s,\\rho_v)$ and $L(s,\\pi_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,\\rho)=L(s,\\pi)$, with $\\rho_v \\leftrightarrow \\pi_v$ being given by the local Langlands correspondence at all $v$. In particular, $\\pi$ is tempered and $L(s,"},"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":"1502.04175","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-14T06:52:14Z","cross_cats_sorted":[],"title_canon_sha256":"42d55b6fed80e6c03cda35c9a6962ad2fc3054cb4ff92b0b10f344ea9aec20af","abstract_canon_sha256":"c51e1d9c88cb326e86a39d553df6b104b8ba0c9f950b292f376429a2cec478c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:49.227855Z","signature_b64":"Eyslgt0MgcV05/Nu/DxolSVIClffcrYYutz7cYBuZditLgOZsrQi7WmoRWfozT9Nphn8gwWBYcuOP4Opq4ksAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5667528ed3792b14878116cc7b5bd8fd4bbf9a66525613a1a9e60e2fcfb675fe","last_reissued_at":"2026-05-18T00:35:49.227113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:49.227113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dinakar Ramakrishnan, Kimball Martin","submitted_at":"2015-02-14T06:52:14Z","abstract_excerpt":"Let $F/k$ be a cyclic extension of number fields of prime degree. Let $\\rho$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $\\pi$ a cuspidal automorphic representation of GL$_2(\\mathbb A_F)$, such that the $L$-functions $L(s,\\rho_v)$ and $L(s,\\pi_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,\\rho)=L(s,\\pi)$, with $\\rho_v \\leftrightarrow \\pi_v$ being given by the local Langlands correspondence at all $v$. 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