Notes on Low Degree L-Data

These notes are an extended version of a talk given by the author at the conference "Analytic Number Theory and Related Areas", held at Research Institute for Mathematical Sciences, Kyoto University in November 2015. We are interested in "$L$-data", an axiomatic framework for $L$-functions introduced by Andrew Booker in 2013. Associated to each $L$-datum, one has a real number invariant known as the degree. Conjecturally the degree $d$ is an integer. Moreover, if $d\in\mathbb{N}$ then one expects that the $L$-datum is that of a $GL_n(\mathbb{A}_F)$-automorphic representation, for some number field $F$. In fact, if $F=\mathbb{Q}$, then $n=d$. This statement was shown to be true for $0\leq d<5/3$ by Booker in his pioneering paper, 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--Perelli in the Selberg class (the best known to date). 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.