We characterize, in terms of the Beurling-Malliavin density, the discrete spectra \Lambda\subset\mathbb R for which a generator exists, that is a function \varphi\in L^1(\mathbb R) such that its \Lambda -translates \varphi(x-\lambda), \lambda\in\Lambda , span L^1(\mathbb R) . It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra \Lambda\subset\mathbb R which do not admit a single generator while they admit a pair of generators.