On the Euler Product of Some Zeta Functions

It is well-known that the Euler product for the Riemann zeta function ζ(s) is still valid for !(s) = 1 and s "= 1. In this paper, we extend this result to zeta functions of number fields. In particular, the Dedekind zeta function ζk(s) for any algebraic number field k and the Hecke zeta function ζ(s,χ) for the rational number field are shown to have the Euler product on the line !(s) = 1 except at s = 1. A functional equation is obtained for the finite Euler product, which is the product of first finite number of factors in the Euler product of ζ(s,χ).