Euler product of an L-function (reviewed)

It is expected that the Euler product of an L-function of degree $d$ and conductor $N$ can be written as $$L(s)=\prod_p L_p(s)$$ where for $p\nmid N$ $$L_p(s)=\prod_{n=1}^d \left( 1-\frac{\alpha_{n}(p)}{p^s}\right)^{-1} \text{ with } |\alpha_{n}(p)|=1$$ and for $p\mid N$, $$L_p(s)=\prod_{n=1}^{d_p}\left( 1-\frac{\beta_{n}(p)}{p^s}\right)^{-1} \text{ where } d_p<d \text{ and } |\beta_n(p)|\le 1.$$ The functions $L_p(s)$ are called Euler factors (or local factors).