The landscape of L-functions: degree 3 and conductor 1

Paper i proceeding, 2024

We extend previous lists by numerically computing approximations to many L-functions of degree d = 3, conductor N = 1, and small
spectral parameters. We sketch how previous arguments extend to say that
for very small spectral parameters there are no such L-functions. Using the
case (d, N) = (3, 1) as a guide, we explain how the set of all L-functions
with any fixed invariants (d, N) can be viewed as a landscape of points in a
(d − 1)-dimensional Euclidean space. We use Plancherel measure to identify
the expected density of points for large spectral parameters for general (d, N).
The points from our data are close to the origin and we find that they have
smaller density.