Let $R_\infty$ be the ring of power series in a single variable with rational coefficients that converge in the whole complex plane. Let $R_\rho$ be the subring of $R_\infty$ that defines holomorphic functions of order no more than $\rho$.

What is $\operatorname{Spec} R_\rho$? Can we describe the vanishing sets of $\sin x$ and other simple power series?

This question was vaguely inspired by the idea of transalgebraic Galois theory.

About all I can say is that if $\rho<\infty$ then irreducible polynomials in $\mathbb Q[x]$ remain irreducible in $R_\rho$, but it is not even obvious to me whether they remain prime ideals.