That is, does there exist a sequence $\{a_n\}_{n\ge 1}\subset\mathbb{C}$, such that $$ \sum_{n\ge 1} \frac{a_n}{n^z} = \sum_{n\ge 1} a_n (z-r)^n $$ for some $r\in\mathbb{C}$? Or perhaps shift there's a shifted version: $$ \sum_{n\ge 1} \frac{a_n}{n^z} = \sum_{n\ge 0} a_{n+1} (z-r)^n $$ Is is possible for a function's Dirichlet series and Taylor series to have the same coefficients?
One can attempt to expand the Taylor series of the Dirichlet series term-by-term, but then you find an infinite system of equations for the coefficients $a_n$, but in general if such systems are not triangular, solving them is difficult, so there might be an easier way.