Dirichlet Series of weakly multiplicative characters

Question

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ \textit{gcd}(m,n) = 1 $). Then the claim is that $$ a(p^{n}) = (\alpha_{p}^{n+1}-\alpha_{p}'^{n+1})/(\alpha_{p}-\alpha_{p}') $$ if $ \alpha_{p} \neq \alpha_{p}' $. My tepmtation is argue $$ \sum_{1}^{\infty}a(n)n^{-s} = \prod_{p}\sum_{n=1}^{\infty}a(p^{n})p^{-ns} $$ and use the expansion of a geometric series. However as the intention is to use this result on Dirichlet series coming from an eigenform the $ a(p)^{n} \neq a(p^{n}) $, and furthermore it is not clear that $ |a(p)p^{-s}|<1 $. If anybody knows a proof or has any suggestions it would be much appreciated.