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Zeros and poles of Artin L-series

  • Richard Foote (a1) and V. Kumar Murty (a2)

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Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

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[1]Curtis, C. and Reiner, I.. Representation Theory of Finite Groups and Associative Algebras (John Wiley and Sons, 1966).
[2]Feit, W.. Characters of Finite Groups (Benjamin, 1967).
[3]Heilbronn, H.. On real zeros of Dedekind ξ-functions. Canad. J. Math. 4 (1973), 870873.
[4]Langlands, R.. Base Change for GL(2). Ann. of Math. Stud. no. 96 (Princeton University Press, 1980).
[5]Stark, H.. Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23 (1974), 135152.

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Zeros and poles of Artin L-series

  • Richard Foote (a1) and V. Kumar Murty (a2)

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