Dedekind Zeta Function - Relations To Other L-functions

Relations To Other L-functions

For the case in which K is an abelian extension of Q, its Dedekind zeta function can be written as a product of Dirichlet L-functions. For example, when K is a quadratic field this shows that the ratio

is the L-function L(s, χ), where χ is a Jacobi symbol as Dirichlet character. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss.

In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G.

Additionally, ζ_K(s) is the Hasse–Weil zeta function of Spec O_K and the motivic L-function of the motive coming from the cohomology of Spec K.