If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor coefficients of $\phi$
Does the following integral converge for $\Re(z)>0$?
$$\int_1^\infty \phi(w)w^{-z-1}\,\mathrm dw.$$
Essentially, I'm wondering if we can bound $\phi$. I'm thinking this is possible if its Taylor coefficients decay sufficiently fast enough. Is there a direct stronger bound on $\phi$ I can find by knowing bounds on its Taylor coefficients?
I'm at a loss observing the growth and convergence of this integral. Any help, literature, hints, comments, insight would be greatly appreciated.