In answer to questions recently raised by Merel [Mer], we prove two non-van
ishing theorems for the central value of automorphic L-functions: let p be
prime and let chi be a primitive character module p. Then for all p large e
nough
1. If chi is not quadratic and even, there exists a primitive weight 2 form
f of level p with L(f x chi, 1/2) not equal 0.
2. If chi is quadratic and even, then there exists a primitive weight 2 for
m f of level p with ord (s=1/2) L(f,s)L(f x chi ,s) = 1.