Northcott property for special values of L-functions.

Pick a integral number n. Consider a family of objects, such that each object $X$ in the family has an L-function $L(X,s)$. If we assume that the collection of special values $L(X,n)$ is bounded, does it imply that the family of objects is finite? We will first explain why we consider this question, and give two recent results. This is joint work with Riccardo Pengo.