I have a Dirichlet series $A(s)$ with an absolutely convergent Euler product for $\sigma >0$. The zeros of the factors converge to $0+2\pi k$. I now have to proof that there can't be an analytic continuation of $A(s)$ to the line $\sigma =0$.
What I thought: If $A(s)$ is analytic at some point $it$, then $\lim_{h\to 0} \frac{1}{h}(A(it+h)-A(it))$ must exist. But $A(it+h)=0$ because of the zeros of the Euler product (is that right?), so $A(it)$ has to be $0$.
Now two questions: 1. Is that right so far? 2. If so, what now? How could I prove that $A(it)=0$ can't be an analytic continuation?