Arithmetic distribution of tempered components of cuspidal representations of GL(3)

Let π=⊗v′πv be an arbitrary unitary cuspidal representation of GL (3) over a number field F. We show, for certain ray classes C, that ∑p2∈C|aπ(p)|<1logN(p)N(p)1/2=+∞,which implies that there are infinitely many unramified places v in C such that π_v’s are tempered with Hecke eigenvalues lying inside the open unit disk. Furthermore, when F= Q, we consider the problem on the least prime p in an arithmetic progression such that π_p satisfies the Ramanujan conjecture. An effective upper bound of Linnik type for such a prime p is proved.