In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta function $\zeta_{K, A}(s) = \prod_{P \in A} (1-||P||^{-s})^{-1}$ can be extended to a meromorphic function on some neighborhood (in the complex plane) of $s=1$ , with the order of pole at $s=1$ equal to $m$, then the polar density is defined to be $m/n$.
My question regarding this definition is: why do we expect the existence of a meromorphic extension when RAISING TO A POWER, in the first place?
Many thanks in advance!