Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and Langlands' philosophy. So far, only a small part of such a theory has been put on a firm basis.

Given ${\displaystyle \rho }$, a representation of ${\displaystyle G}$ on a finite-dimensional complex vector space ${\displaystyle V}$, where ${\displaystyle G}$ is the Galois group of the finite extension ${\displaystyle L/K}$ of number fields, the Artin ${\displaystyle L}$-function: ${\displaystyle L(\rho ,s)}$ is defined by an Euler product. For each prime ideal ${\displaystyle {\mathfrak {p}}}$ in ${\displaystyle K}$'s ring of integers, there is an Euler factor, which is easiest to define in the case where ${\displaystyle {\mathfrak {p}}}$ is unramified in ${\displaystyle L}$ (true for almost all ${\displaystyle {\mathfrak {p}}}$). In that case, the Frobenius element ${\displaystyle \mathbf {Frob} ({\mathfrak {p}})}$ is defined as a conjugacy class in ${\displaystyle G}$. Therefore the characteristic polynomial of ${\displaystyle \rho (\mathbf {Frob} ({\mathfrak {p}}))}$ is well-defined. The Euler factor for ${\displaystyle {\mathfrak {p}}}$ is a slight modification of the characteristic polynomial, equally well-defined,

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