Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form.

Let $f(s,g)$ be a two variable smooth good (in a suitable sense) function and let $F(s)=\int_{G}f(s,g)dg$. Assume $F(s)$ is absolutely convergent for $\Re(s)>0$ and has meromorphic continuation to all $s \in C$(complex number). Also we know that $$\lim_{s\to 0} s^{m}\cdot F(s)$$ exist for some positive integer $m>0$. Then can we say that $$\lim_{s\to 0} s^{m}\cdot \int_{G}|f(s,g)|dg$$ exist? Or more weakly, can we ensure that $$\lim_{s\to 0} s^{m+1}\cdot \int_{G}|f(s,g)|dg=0?$$

If it does not holds, would you suggest some mild assumption that compels this to hold? In my case, I am considering only when $F(s)$ is given as a local-zeta integral from some tempered representation and some degenerate principal series representation. Then the local-zeta integral has the above properties. If it does not hold in general, then does it hold for local-zeta integral?

Any help or comments will be greatly appreciated!