Self-dual Artin Representations
نویسنده
چکیده
Functional equations in number theory are relations between an L-function and some sort of dual L-function, and in general, the L-function and its dual need not coincide. For example, if χ is a primitive Dirichlet character then the functional equation relates L(s, χ) to L(1− s, χ), and L(s, χ) = L(s, χ) if and only if χ = 1. Or if f is a primitive cusp form of weight two for Γ1(N) and f∨ is the complexconjugate form then the functional equation relates L(s, f) to L(2 − s, f∨), and L(s, f∨) = L(s, f) if and only if f is a cusp form for Γ0(N) with trivial character. Let us call an L-function self-dual if its functional equation is a relation between the L-function and itself. While self-dual L-functions are often of special interest, the preceding examples suggest that they may also be rare. Indeed the number of Dirichlet characters modulo N is the quantity
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