About the values of the $\Gamma$ function

Question

The $\Gamma$ function is defined by

$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$

where $z$ is a complex number.

We know that if $z$ is real then the values of $\Gamma$ are also real. I am interested in the case where $z$ is complex and the values of $\Gamma$ are real. In particular where $0<Re(z)<1$.

Is this case possible?

Answer 1

Here's a picture of the contours for $\Im(\Gamma(z))=0$. As you can see, pure real values of the gamma function, and indeed most analytic functions in the complex plane, are quite common off the real line.