Let $F$ be an L-function in the sense of the Selberg class --> http://en.wikipedia.org/wiki/Selberg_class.
We are observing the integral $\frac{1}{2\pi i}\int_{(c)}F(s+w)\Gamma(w)z^w dw$ for $\Re(s)>1$ and $c>0$. Why can we shift the line of integration to the left, say, to $(-R)$, where $R$ is some positive real number, non-integer, the difference coming only from the residues in between?
This is equivalent to asking why does the integral of $F(s+w)\Gamma(w)z^w$ over the horizontal line segments vanish when we push them to $\pm i\infty$?
The thing is that by doing so we are moving the line of integration over the critical stripe of $F$. Outside the critical stripe we can combine the Stirling's approximation of $\Gamma(w)$ with the fact that $F(\sigma+ it)$ is $o(t)$, when $t\to\infty$, as a Dirichlet series, plus Stirling's approximation for the Gamma-factors of the functional equation, when we observe the asymptotics of $F$ left from the critical stripe. However, is there a simple way to obtain some basic "vertical asymptotics" of $F$ over the critical stripe that would allow us the aforementioned move?
My only idea is to use the fact that $(s-1)^m F(s)$ ($m$ is the multiplicity of the pole of $F$ in $s=1$) is an entire function of finite order. But unless the order is $1$, it would actually appear to overweight any Stirling's approximation. I guess, I am missing something obvious in the whole story.
The reason why I am asking this is because the above argument appears to be pretty standard throughout various materials conerning L-functions, starting with the more elementary Dedekind-L-functions and going through automorphic L-functions and similar. Thus is seems to be a general argument that is not closely related to the specifics of each of these L-functions, and I would like to understand the principle behind it. To be honest, it has been bugging me for a few weeks now...
Thanks in advance for any help!