Let $F(\sigma+it)=\sum_{n=1}^{\infty}a_n n^{-(\sigma+it)}$ a Dirichlet series that is absolutely convergent in the half-plane $\sigma>\sigma_a$ and let $G(\sigma+it)$ be a function such that $$F(\sigma)=G(\sigma),$$ for all $\sigma>\sigma_a$.
In this general setting is it true that $F(\sigma+it)=G(\sigma+it)$, in the half-plane $\sigma>\sigma_a$?