L-functions from nothing

I will report on joint work in progress with Andrew Booker on the practical implementation of an axiomatic approach to the enumeration of arithmetic L-functions that lie in a certain subset of the Selberg class that is expected to include all L-functions of abelian varieties. As in the work of Farmer, Koutsoliotas, and Lemurell, our approach is based on the approximate functional equation. We obtain additional constraints by considering twists (and more general Rankin-Selberg convolutions) of our unknown L-function that yield a system of linear constraints that can be solved using the simplex method. This allows us to significantly extend the range of our computations for the family of L-functions associated to abelian surfaces over ℚ. We also introduce a method for certifying the completeness of our enumeration.