Notes on low degree L-data

These notes are an extended version of a talk given by the author at the confer‐
ence Analytic Number Theory and Related Areas held at Research Institute for Math‐
ematical Sciences, Kyoto University in November 2015. We are interested in L‐data, an
axiomatic framework for L\sim‐functions introduced by Andrew Booker in 2013 [3]. Associated
to each L‐datum, one has a real number invariant known as the degree. Conjecturally the
degree d is an integer, and if d\in \mathrm{N} then the L‐datum is that of a \mathrm{G}\mathrm{L}_{n}(\mathrm{A}_{F}) ‐automorphic
representation for n\in \mathrm{N} and a number field F (if F=\mathbb{Q} ,
then n=d This statement was
shown to be true for 0\displaystyle \leq d<\frac{5}{3} by Booker in his pioneering paper [3], and in these notes we
consider an extension of his methods to 0\leq d<2. This is simultaneously a generalisation
of Booker’s result and the results and techniques of Kaczorowski‐Pereli in the Selberg class
[10]. Furthermore, we consider applications to zeros of automorphic L-‐functions. In these
notes we review Booker’s results and announce new ones to appear elsewhere shortly