  We study the equivariant local epsilon constant conjecture, denoted by
$C_{EP}^{na}(N/K,V)$, as formulated in various forms by Kato, Benois and
Berger, Fukaya and Kato and others, for certain 1-dimensional twists
$T=\mathbb{Z}_p(\chi^{nr})(1)$ of $\mathbb{Z}_p(1)$. Following ideas of recent
work of Izychev and Venjakob we prove that for $T=\mathbb{Z}_p(1)$ a conjecture
of Breuning is equivalent to $C_{EP}^{na}(N/K,V)$. As our main result we show
the validity of $C_{EP}^{na}(N/K,V)$ for certain wildly and weakly ramified
abelian extensions $N/K$. A crucial step in the proof is the construction of an
explicit representative of $R\Gamma(N,T)$.
