This paper is a slightly expanded version of the talk given at the conference. We sketch an example of a non-trivial element of $K_2$ of a certain threefold, whose existence is related to the vanishing of an incomplete $L$-function of a modular form at $s = 1$. This is preceded with a simple account, for the non-specialist, of some of the conjectures (mostly due to Beilinson) which relate ranks of $K$-groups and orders of $L$-functions, supplemented by examples coming from modular forms.