On the minimal norm of a non-regular generalized character of an arbitrary finite group

We prove that for any finite group G, the sum across non-identity elements of the squared absolute value of any generalized character of G which does not vanish on all non-identity elements of G is at least |G|/d -1, where d is the maximal degree of a complex irreducible character of G, and we identify all cases where this minimum possible value is attained.