Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?