Abstract:

In number theory, it is indispensable to study analytic properties of L-functions as they often relate to important problems in this realm. Perhaps two of the most famous problems in this aspect are the Riemann hypothesis and the Artin conjecture. In the first part of this talk, we will introduce L-functions and Maass forms, and we will discuss some basic properties of these two objects. In the second part of this talk, we will present some explicit formulae which relate certain period integrals of Maass forms and the central values of the twisted triple product L-functions. As an application of these formulae and the standard analytic properties of the L-functions, we prove the optimal upper bound of a sum of restricted 2-norms of the -normalized Maass forms on certain quadratic extensions with prime level and bounded spectral parameter.