The completed Riemann zeta function is traditionally defined as $$\xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{s}{2} \right) \zeta(s)$$
I understand the reasons of the presence of Euler gamma and of the power of $\pi$, as well as the two poles at $s=0$ and $s=1$, but... is there a reason for the constant factor $\frac{1}{2}$? It does not change the poles, the functional equation, or any relevant analytic property of $\xi$, right? Is there a deep normalization for it to be present?
It is historical. Take a look at the original Riemann's paper. Instead of $\Gamma(s)$, he uses $\Pi(s)=\Gamma(s+1)=s\Gamma(s)$. This is because he apparently prefers Gauss's defition $\Pi$ (which coincides with factorial at integer points) to $\Gamma$. Thus for him $$\xi(s) = (s-1) \pi^{-s/2} \Pi\left(\tfrac{s}{2} \right) \zeta(s)$$ which is kind of natural. Now, the factor $\frac12$ emerges when you switch to more modern definition $\Gamma$.