Zeros of L-functions have been extensively studied, due to their close connection to arithmetic problems. Despite several precise conjectures about their behavior, our unconditional understanding of them remains limited. In this talk we will discuss certain intrinsic properties of such zeros, focusing on what is known (in degrees one and two) about their accumulation on the central line and their multiplicity. Here the tools of analytic number theory can give quantitative advances, and we will show how to deduce that there are many zeros of multiplicity one for the L-function associated to a modular form.