Integrals of periodic functions describe a wide range of phenomena; for example, the techniques of Fourier analysis associate to suitable period integrals many arithmetically interesting objects. The functions we study in this talk are automorphic forms (arising from classical modular forms). We shall explain how they can be analysed via the theory of representations of groups over local fields. From this perspective we consider the role of a suitably chosen period integral and prove an explicit formula relating it to the special value of an associated L-function. We shall comment on the variety of applications of such a formula to both analytic and algebraic number theory.