Title: First Non-Vanishing Theorems

Abstract: Dirichlet proved that there are primes in arithmetic
progressions, but how big is the first? Given a prime, how large is the
first quadratic non-residue? How many Fourier coefficients determine an
automorphic form? In joint with Jeff Hoffstein we ask the following
analogous question: for an automorphic form f on GL(r), what is the
least fundamental discriminant d such that the L-function associated to
f, twisted by the quadratic character chi_d, does not vanish at the
center of the critical strip? Our partial answer in the cases r=1,2,3
sheds some light on the similarities and differences between using
classical methods and multiple Dirichlet series.?