I asked this question in SEM but I got no answer, so I'm trying my luck here.

Let the Dirichlet series $\phi(s)=\sum_{n\ge 1}\frac{a(n)}{n^s}$ be absolutely convergent for $\Re(s)>1$ and extend to a meromorphic function on the half plane $\Re(s)>1/2$ with only pole at $s=1.$ write $$\phi(s)=\sum_{\text{prime}}\frac{a(p)}{p^s}+\sum_{\text{non-prime}}\frac{a(n)}{n^s}$$ If we suppose that $\sum_{\text{prime}}\frac{a(p)}{p^s}$ converges absolutely for $\Re(s)>1$ and has a pole at $s=1.$ We can deduce that the series $\sum_{\text{prime}}\frac{a(p)}{p^s}$ can be continued analytically to a meromorphic function in the half plane $\Re(s)>1/2$ with only pole at $s=1$ ?