Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for $s$ over $A$ to be $f \in D$ such that $\deg(f)$ is minimal and $f(s) = 0$, where $\deg(f)$ is the first coefficient in the series $f = \sum_{n=1}^{\infty} f_n n^{-X}$, that is nonzero (ie. lowest $n$ st. $f_n \neq 0$).
Then is such a minimal degree interesting or is it always $1$? I know that no maximal exist as you can always multiply the zero-valued series by $\frac{1}{m^{-s}}, \ m \gt 1$, usually.
Thanks.