Consider the following function :
$$f(z)=\frac{e^{az}}{\Gamma(z)\zeta(bz)}$$
$z=x+iy$
I want to know if $|f(z)|$ is bounded by $c/|z|^{1+\epsilon}$?
Where , $a,b,c;\epsilon > 0 $ are constants .
Also , It would be nice if we can tell the following limit exists or not ( and if exists the estimation ):
$$\frac{ |f(z)|}{e^y}$$ as $y$ tends to $\infty$