Classification of Low Degree L-Data

L-data were introduced by A. Booker as an axiomatic framework for L-functions incorporating not only the Selberg class but many other L-functions for which the Ramanajuan conjecture is not yet known (eg. Artin, automorphic). Study of the class of L-data offers immediate applications to the orders of zeros of automorphic L-functions, which are not so clear from the viewpoint of the Selberg class. We say an L-data is "positive" when it resembles an L-function having only finitely many poles. Associated to an L-datum one has a real number invariant known as the degree, which is expected to be integral. In this paper we classify all positive L-data of degree d\in[0,2), generalising the analogous theorem of Kackzorowski-Perelli in the Selberg class, which is the best known to date.