Langlands L functions for groups over finite fields.

Question

In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple Google search isn't helping that much. Say I have a representation $\pi$ of $\mathrm{GL}_2(\mathbb{F}_p)$, for example, if $\pi$ is the principal series of $GL(2,\mathbb{F}_p)$ What does it mean to attach a "Langlands" L-function to this data? What if I add in the data of $\rho$ a rep of the dual group over $\mathbb{C}$. Are they related to some Zeta functions of varieties over finite fields? Do they have functional equations, etc? Any explanation or references would be helpful.