# Artin L-function

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Let K/k be Galois extension of global fields, and let ${\displaystyle \rho }$ be a representation of the Galois group ${\displaystyle \scriptstyle G=\mathrm {Gal} (K/k)}$ on a finite dimensional complex vector space V. The Artin L-function associated to ${\displaystyle \rho }$ is defined by the Euler product
${\displaystyle L(K/k,\rho ,s)=\prod _{\mathfrak {p}}{\frac {1}{\det \left(1-\varphi _{\mathfrak {P}}{\mathfrak {N}}\left({\mathfrak {p}}\right)^{-s};V^{I_{\mathfrak {P}}}\right)}}.}$
The product extends over the set of prime ideals of k, and ${\displaystyle {\mathfrak {P}}}$ is an arbitrarily chosen prime ideal of K dividing ${\displaystyle {\mathfrak {p}}}$. Also, ${\displaystyle \varphi _{\mathfrak {P}}}$ is the Frobenius automorphism in G associated to ${\displaystyle {\mathfrak {P}}}$, and ${\displaystyle I_{\mathfrak {P}}}$ is the corresponding inertial group. The determinant in the definition is independent of the choice of the prime ideal ${\displaystyle {\mathfrak {P}}}$. Also, although the Frobenius automorphism is only determined up to an element of ${\displaystyle I_{\mathfrak {P}}}$, the determinant is independent of this choice.