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 [Boo15]. 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 ∈ N then the L-datum is associated to a GL_n(A_F )- automorphic representation for n ∈ N and a number field F (if F = Q and there is no scaling, then n = d.). This statement was shown to be true for 0 ≤ d < 5/3 by Booker in his pioneering paper [Boo15], and in these notes we consider an extension of his methods to 0 ≤ d < 2. This is simultaneously a generalisation of Booker’s result and the results and techniques of Kaczorowski–Perelli in the Selberg class [KP11]. 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 [Oli16].