Editing Dedekind zeta function (section)

==Definition and basic properties==
Let ''K'' be an [[algebraic number field]]. Its Dedekind zeta function is first defined for [[complex number]]s ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1 by the Dirichlet series

:<math>\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}</math>

where ''I'' ranges through the non-zero [[ideal (ring theory)|ideals]] of the [[ring of integers]] ''O''<sub>''K''</sub> of ''K'' and ''N''<sub>''K''/'''Q'''</sub>(''I'') denotes the [[absolute norm]] of ''I'' (which is equal to both the [[Index of a subgroup|index]] [''O''<sub>''K''</sub>&nbsp;:&nbsp;''I''] of ''I'' in ''O''<sub>''K''</sub> or equivalently the [[cardinality]] of the [[quotient ring]] ''O''<sub>''K''</sub>&nbsp;/&nbsp;''I''). This sum converges absolutely for all complex numbers ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1. In the case ''K''&nbsp;=&nbsp;'''Q''', this definition reduces to that of the Riemann zeta function.

===Euler product===
The Dedekind zeta function of <math>K</math> has an Euler product which is a product over all the non-zero [[prime ideal]]s <math>\mathfrak{p}</math> of <math>\mathcal{O}_K</math>

:<math>\zeta_K (s) = \prod_{\mathfrak{p} \subseteq \mathcal{O}_K} \frac{1}{1-N_{K/\mathbf{Q}}(\mathfrak{p})^{-s}},\text{ for Re}(s)>1.</math>

This is the expression in analytic terms of the [[Dedekind domain|uniqueness of prime factorization of ideals]] in <math>\mathcal{O}_K</math>. For <math>\mathrm{Re}(s)>1,\ \zeta_K(s)</math> is non-zero.

===Analytic continuation and functional equation===
[[Erich Hecke]] first proved that ''ζ''<sub>''K''</sub>(''s'') has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole at ''s''&nbsp;=&nbsp;1. The [[Residue (complex analysis)|residue]] at that pole is given by the [[analytic class number formula]] and is made up of important arithmetic data involving invariants of the [[unit group]] and [[class group]] of ''K''.

The Dedekind zeta function satisfies a functional equation relating its values at ''s'' and 1&nbsp;&minus;&nbsp;''s''. Specifically, let Δ<sub>''K''</sub> denote the [[Discriminant of an algebraic number field|discriminant]] of ''K'', let ''r''<sub>1</sub> (resp. ''r''<sub>2</sub>) denote the number of real [[Algebraic number theory#Primes and places|places]] (resp. complex places) of ''K'', and let

:<math>\Gamma_\mathbf{R}(s)=\pi^{-s/2}\Gamma(s/2)</math>

and

:<math>\Gamma_\mathbf{C}(s)=
(2\pi)^{-s}\Gamma(s)</math>

where Γ(''s'') is the [[gamma function]]. Then, the functions

:<math>\Lambda_K(s)=\left|\Delta_K\right|^{s/2}\Gamma_\mathbf{R}(s)^{r_1}\Gamma_\mathbf{C}(s)^{r_2}\zeta_K(s)\qquad
\Xi_K(s)=\tfrac12(s^2+\tfrac14)\Lambda_K(\tfrac12+is)
</math>

satisfy the functional equation

:<math>\Lambda_K(s)=\Lambda_K(1-s).\qquad
\Xi_K(-s)=\Xi_K(s)\;</math>