## Specific Case

The periodicity is obvious from computation: $$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$ However, is there a way to see directly from the integral that it should be periodic in $x$ with period $2\pi i$?

One thing you can read off of the integral is that it satisfies a differential equation, although I don't see why this would give periodicity.

One way to tackle this could be by using a $\beta$-integral to prove multiplicativity of the function

## Generalization

The above function naturally lives on $\mathbb{C}/(2\pi i\mathbb{Z})$.

What about $\cal{M}^{-1}\{\prod_{i=1}^n\Gamma(\lambda_i s+\mu_i)\}$ for some $\lambda_i>0,\mu_i\in\mathbb{C}$. In certain cases this function seems to naturally live on a quotient of a Siegel space or on a symmetric space. Is there some general result about this?