Zeroes of p-adic L-functions: the Ellenberg-Jain-Venkatesh conjecture

We introduce p-adic L-functions and their main properties using Mazur's contruction and p-adic measures; we also discuss what is known and what is expected about the distribution of zeroes for both complex and p-adic L-functions. After this, we focus on the study of the Elienberg-Jain-Venkatesh conjecture (2011) about the zeroes of family of p-adic L-functions corresponding to imaginary quadratic fields; we support this study also by doing some numerical experiments on the lambda invariants and on the order of these zeroes.