Title: Extremal invariant polynomials not satisfying the Riemann hypothesis,
Abstract: Zeta functions for linear codes were defined by Iwan Duursma in 1999. They
were generalized to the case of some invariant polynomials by the preset
author. One of the most important problems is whether extremal weight
enumerators satisfy the Riemann hypothesis. In this article, we show there
exist extremal polynomials of the weight enumerator type which are invariant
under the MacWilliams transform and do not satisfy the Riemann hypothesis.