  The density conjecture of Katz and Sarnak predicts that, for natural families
of L-functions, the distribution of zeros lying near the real axis is governed
by a group of symmetry. In the case of the universal family of automorphic
forms of bounded analytic conductor on a totally definite quaternion algebra,
we determine the associated distribution for a restricted class of test
functions. In particular it leads to non-trivial results on densities of
non-vanishing at the central point.
