# Collected papers. Vol.3 (1964-1978) by Weil A.

By Weil A.

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30 E. Kowalski The best knpwn results-proved using the spectral theory of automorphic forms (see Chapter 3)-for primes in arithmetic progressions go beyond what is immediately provable from GRH (Bombieri-Friedlander-Iwaniec); they are commonly used in applications. The same results and difficulties occur in the Chebotarev density theorem for Artin £-functions, often exacerbated because the degree of interesting families of fields is larger than that of cyclotomic fields, making even the form of the prime ideal theorem based on GRH insufficient for applicat10ns.

Since the inverse of the gamma factor is known to be an entire function of order = 1, it follows that L(x, s) is also. This is actually very far from the truth. Of course, for Re(s) > 1, where the series converges absolutely, it follows that L(x, s) « I, uniformly for Re(s) > 1 + 8, 8 > 0, for any x, the implied constant depending only on K. For Re(s) < 0, on the other hand, this implies, thanks to the functional equation, Stirling's formula and the 8Multiply by s(l - s) if X = I. 2. Elementary Theory of L-Functions II 33 shape of the gamma factor, that L(x,s) « lsl(l-a)/ 2 , uniformly fora =Re(s) < -8, 8 > 0.

Definition. : 1 an integer and x a Dirichlet character modulo q. :if = Af and f I y = x(a)f, for y E ro(q). 1. Taking y = -1, on gets the relation f = f lk y = x ( -1)( -1)k f, so there can exist nonzero holomorphic modular forms only if the character satisfies the consistency condition x (-1) = ( -1 )k, which is tacitly assumed to be the case in what follows. Similarly for automorphic functions, we must have x (-1) = 1. 3. Classical Automorphic Forms 47 One can also define nonholomorphic forms of weight k =F 0, using a modified differential operator.