= Artin L-function
{c}
{title2=$L(s,\rho)$}
For a finite <Galois extension> $L/K$ of <number fields> and a finite-dimensional complex <group representation> $\rho$ of its <Galois group>, the finite part of its <Artin L-function> is the <Euler product>
$$
L(s,\rho)=\prod_{\mathfrak p}\det\!\left(1-(N\mathfrak p)^{-s}\rho(\operatorname{Frob}_{\mathfrak p})\bigm|V^{I_{\mathfrak p}}\right)^{-1},\qquad\operatorname{Re}s>1.
$$
Here $I_{\mathfrak p}$ is the <inertia group>, and a <Frobenius automorphism> lift acts on its invariant subspace. Different lifts induce the same operator there. At an unramified prime the invariant subspace is all of $V$; at a ramified prime this restriction removes the part on which inertia acts nontrivially.
The trivial representation gives the <Dedekind zeta function> of $K$. Direct sums of representations give products of <Artin L-functions>, because <determinants> multiply on direct sums. Taking the logarithm of each local factor produces traces of powers of Frobenius, connecting representation-theoretic characters with prime-counting sums. Archimedean factors are added separately when forming a completed L-function.
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