Sharp characters of finite groups

If χ is a virtual (generalized) character of a finite group G, with n=χ and L={χ(g)|g∈G, g≠1 {χ(g)|gε{lunate}G, g ≠ 1}, then |G| dividesfL(n), where fL(x) is the monic polynomial of least degree having L as its set of roots. (This generalises a result of the second author for permutation characters.) We say that the pair ((G,χ)) is L-sharp if |G|=fL(n). We characterise the L-sharp pairs for various sets L, sometimes under additional hypotheses, and give a number of examples.