Chenevier and Lannes proved that there are only finitely many level 1 algebraic automorphic representations of PGL_n of motivic weight ≤23. Furthermore, they completely classified 11 such representations of motivic weight ≤22. The proof is based on the explicit formula for the Jacquet–Piatetski-Shapiro–Shalika L-functions. Positivity results impose restrictions on the L-parameters of the archimedean components of such representations.