For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into $$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$

where $\chi_N$ is the primitive character modulo $N$. Now take $\pi$ to be an automorphic representation for $GL_2$. What can be said about the local L-factor associated to $\pi$? Do we know the specific form in which it can be written? (something like the same but with $N$ being the depth, or the conductor of $\pi$?)