From my previous question, I know that$$\zeta_X(s) = {{P(u)}\over{(1-u)(1-au)}}$$for some polynomial $P(u)$ of degree $\le 2g$, where

• $X$ is the set of all places of $F$, a function field in one variable over a total constant field $k$,
• $u := q^{-s}$,
• $g$ is the genus of $F$.

My question now is, must the limit$$\lim_{s \to 1} (1 - q^{1-s})\zeta_X(s)$$necessarily converge to$${h\over{(q-1)q^{g-1}}}?$$Here, $h$ denotes the number of divisor classes of $F$ of degree $0$.