Extension of an identity related to a Dirichlet series

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 $$\sigma>\sigma_a$$?

• What do a know about $G$? – Kavi Rama Murthy Apr 4 at 10:29
• @KaviRamaMurthy it is a product of a Dirichlet series with an integral depending on this same DS. – GoldSoundz Apr 4 at 10:33