Analytic continuation of Dirichlet series of arithmetic function

I am trying to prove that the Dirichlet series of a multiplicative function with polynomial growth that vanishes at all but finitely many primes admits analytic continuation to $ Re(s)> 1/2$. Using the convergence criterium for infinite products this reduces to showing that for any $\sigma >1/2$ the quantity $\sum_{n\ge 2} \sum_{p} n^A p^{-\sigma n}$ is finite.(here $p$ ranges over primes and $A $ is a non negative integer). However I do not know how to show finiteness of the sum. Any help is appreciated.