one of the criteria for $L(s,f)$ to be an $L$-function is if it has a Gamma factor
$\gamma(f,s)=\pi^{-ds/2} \prod_{j=1}^d \Gamma\left(\frac{s+k_j}{2}\right)$ where d is the degree of the Euler product. However, I do not understand how one determines the value of $k_j$.
For instance, for the Dedekind zeta function, we have $\gamma(f,s)=\pi^{-ds/2} \Gamma\left(\frac{s}{2}\right)^{r_1+r_2} \Gamma\left(\frac{s+1}{2}\right)^{r_2}$ where $r_1$ is the number of real embeddings of the field K and $r_2$ the number of pairs of complex embeddings such that $d=r_1+2r_2$.
I do not understand why $k_j= 0$ or $1$.
Thanks