In what way do the Weil Conjectures pertain to Langlands?

Question

For a relative variety $X$ over a ring of integers $O_K$, we can define a zeta function. This zeta function is defined as the product of the zeta functions of the variety specialized to $O_K/\mathfrak{p}$ as $\mathfrak{p}$ runs over $Spec(O_K)$. In turn, zeta functions of varieties over finite fields are easy to define using the counting of rational points. As Grothendieck proved, these zeta functions can be expressed as a product of $L$-functions indexed by $i$, where the $i^{th}$ $L$-function is related to the $i^{th}$ (Weil) cohomology of $X_{O_K/\mathfrak{p}}$. The $i^{th}$ $L$-function of $X$ over $O_K$ is defined to be the product over $\mathfrak{p}$ of the $i^{th}$ $L$-function of $X_{O_K/\mathfrak{p}}$.

The Weil conjectures give us a lot of information about the zeta functions of varieties over finite fields, and in fact about their $L$-functions.

The Langlands program is about properties of $L$-functions of $X$ over $O_K$.

Is it possible to interpret the Weil conjectures as telling us something meaningful about the Langlands program?

Answer 1

If $\pi$ is a regular algebraic cuspidal automorphic representation of $GL_n/K$ with K totally real or CM, and $\pi$ satisfies a certain self-duality condition, then $\pi_v$ is tempered for all finite $v$. This monumental theorem is a vast generalization of Deligne's proof of the Ramanujan conjecture, and the proof ultimately appeals to the Weil conjectures, by proving an identity $L(s,\pi)=L(s,M(\pi))$ where $M(\pi)$ is (essentially) a submotive of the cohomology of a Shimura variety.