Generalized Ramanujan–Selberg conjecture for unramified cuspidal representations of GL(n)

Let $\pi$ be a cuspidal automorphic representation of $GL(n)$ that is unramified at a prime $p$, and let $\alpha_{p,j}$ be the quantities appearing in its local Euler product. Generalized Ramanujan–Selberg conjecture.

$$|\alpha_{p,j}|=1$$

for every relevant $j$. This extends the Ramanujan and Selberg conjectures through the assertion that the unramified local parameters are tempered; the conjecture remains open in general.