Unfaithful minimal Heilbronn characters of finite groups

We obtain necessary and sufficient conditions for a finite group G to possess an “unfaithful minimal Heilbronn character”—a virtual character but not a character of G whose inner product with every monomial character is nonnegative, whose restriction to every proper subgroup and quotient is a character, and whose restriction to some proper subgroup is unfaithful. We give an application constraining hypothetical minimal counterexamples to Artin's Conjecture on the holomorphy of L-series.