Functional Equation
Artin L-functions satisfy a functional equation. The function L(s, ρ) is related in its values to L(1 − s, ρ*), where ρ* denotes the complex conjugate representation. More precisely L is replaced by Λ(s, ρ), which is L multiplied by certain gamma factors, and then there is an equation of meromorphic functions
- Λ(s, ρ) = W(ρ)Λ(1 − s, ρ*)
with a certain complex number W(ρ) of absolute value 1. It is the Artin root number. It has been studied deeply with respect to two types of properties. Firstly Langlands and Deligne established a factorisation into Langlands–Deligne local constants; this is significant in relation to conjectural relationships to automorphic representations. Also the case of ρ and ρ* being equivalent representations is exactly the one in which the functional equation has the same L-function on each side. It is, algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is, then, either +1 or −1. The question of which sign occurs is linked to Galois module theory (Perlis 2001).
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