Kaczorowski’s lectures present a survey of the axiomatic class S of Lfunctions
introduced by Selberg. Essentially, the main aim of the Selberg
class theory is to prove that such an axiomatic class coincides with the class
of automorphic L-functions. Although the theory is rich in interesting conjectures,
the focus of these lecture notes is mainly on unconditional results. After
a chapter on classical examples of L-functions and one on the basic theory,
the notes present an account of the invariant theory for S. The core of the
theory begins with chapter 4, where the necessary material on hypergeometric
functions is collected. Such results are applied in the following chapters, thus
obtaining information on the linear and non-linear twists which, in turn, yield
a complete characterization of the degree 1 functions and the non-existence
of functions with degree between 1 and 5/3.