  The Riemann Zeta-Function is the most studied L-function; it's zeroes give
information about the prime numbers. We can associate L-functions to a wide
array of objects, and in general, the zeroes of these L-functions give
information about those objects. For arbitrary L-functions, the order of
vanishing at the central point is of particular important. For example, the
Birch and Swinnerton-Dyer conjecture states that the order of vanishing at the
central point of an elliptic curve L-function is the rank of the Mordell-Weil
group of that elliptic curve.
  The Katz-Sarnak Density Conjecture states that this order vanishing and other
behavior are well-modeled by random matrices drawn from the classical compact
groups. In particular, the conjecture states that an average order vanishing
over a family of L-functions can be bounded using only a given weight function
and a chosen test function, phi. The conjecture is known for many families when
the test functions are suitably restricted.
  It is natural to ask which test function is best for each family and for each
set of natural restrictions on phi. Our main result is a reduction of an
otherwise infinite dimensional optimization to a finite-dimensional
optimization problem for all families and all sets of restrictions. We
explicitly solve many of these optimization problems and compute the improved
bound we obtain on average rank. While we do not verify the density conjecture
for these new, looser restrictions, with this project, we are able to precisely
quantify the benefits of such efforts with respect to average rank. Finally, we
are able to show that this bound strictly improves as we increase support.
