Editing Dedekind zeta function (section)

==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-function]]s. For example, when ''K'' is a [[quadratic field]] this shows that the ratio

:<math>\frac{\zeta_K(s)}{\zeta_{\mathbf{Q}}(s)}</math>

is the ''L''-function ''L''(''s'',&nbsp;χ), where χ is a [[Jacobi symbol]] used 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|Artin ''L''-function]] of the [[regular representation]] of ''G'' and hence has a factorization in terms of Artin ''L''-functions of [[irreducible representation|irreducible]] [[Artin representation]]s of ''G''.

The relation with Artin L-functions shows that if ''L''/''K'' is a Galois extension then <math>\frac{\zeta_L(s)}{\zeta_K(s)}</math> is holomorphic (<math>\zeta_K(s)</math> "divides" <math>\zeta_L(s)</math>): for general extensions the result would follow from the [[Artin conjecture (L-functions)|Artin conjecture for L-functions]].<ref name=Mar19>Martinet (1977) p.19</ref>

Additionally, ''ζ''<sub>''K''</sub>(''s'') is the [[Hasse–Weil zeta function]] of [[Spectrum of a ring|Spec]] ''O''<sub>''K''</sub><ref>{{harvnb|Deninger|1994|loc=§1}}</ref> and the [[motivic L-function|motivic ''L''-function]] of the [[motive (algebraic geometry)|motive]] coming from the [[cohomology]] of Spec ''K''.<ref>{{harvnb|Flach|2004|loc=§1.1}}</ref>