Artin L-function

Definition
Let $$L/K$$ be a Galois extension of fields, and $$G$$ its Galois group. Let $$\rho$$be a linear representation of $$G$$ over $$\mathbb{C}$$. (In other words, $$\rho$$ is a Galois representation over the complex numbers).

The Artin L-function associated with $$\rho$$, denoted as $$s \mapsto L(\rho,s)$$, is defined as follows: it is the product, over all prime ideals $$P$$, of the following Euler factor corresponding to that $$P$$:

$$det(1 - t\rho(Frob(P)))^{-1}$$

evaluated at $$t = N(P^{-s})$$.

Strictly speaking, the above definition works when $$P$$ is unramified. A slight variant works when $$P$$ is ramified.

For Abelian Galois group
When the underlying Galois group $$G$$ is Abelian, the Artin L-function specializes to the Hecke L-function.

For Abelian Galois group and over rationals
When $$G$$ is Abelian and $$K = \mathbb{Q}$$, the Artin L-function specializes to the Dirichlet L-function.