Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Zeros and poles of Artin L-series

Richard Footea1 and V. Kumar Murtya2 p1

a1 Department of Mathematics, University of Vermont, Burlington, VT 05405, U.S.A.

a2 Department of Mathematics, Concordia University, Montréal, H3G 1M8, Canada

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 xs3008χ, 1xs3009, 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.

(Received November 24 1987)

(Revised June 14 1988)

Correspondence:

p1 Current address: Department of Mathematics, University of Toronto, Toronto, Canada, M5S 1A1.