Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents information about the ideals in the corresponding number ring, generalizing how the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} represents information about the factorization of integers. Dedekind zeta functions generalize many properties of the Riemann zeta function: they can be defined as a Dirichlet series, have an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, have an Euler product expansion, or satisfy a functional equation.