The theory of general Dirichlet series and the theory of power theory have some analogs:
The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series.(wiki/General_Dirichlet_series)
By the Riemanian mapping theorem, the open unit disc $D$ and open upper half-plane $H$ are conformally equivalent. So I wonder if there is a bijective conformal map $\phi:D\to H$ such that any general Dirichlet series on $H$ is pulled back by $\phi$ into a power series on $D$.
A related question is reduction-of-dirichlet-series-into-power-series.