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We study the equivariant local epsilon constant conjecture, denoted by$C_{EP}^{na}(N/K,V)$, as formulated in various forms by Kato, Benois andBerger, 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 recentwork of Izychev and Venjakob we prove that for $T=\mathbb{Z}_p(1)$ a conjectureof Breuning is equivalent to $C_{EP}^{na}(N/K,V)$. As our main result we showthe validity of $C_{EP}^{na}(N/K,V)$ for certain wildly and weakly ramifiedabelian extensions $N/K$. A crucial step in the proof is the construction of anexplicit representative of $R\Gamma(N,T)$.

The equivariant local $\epsilon$-constant conjecture for unramified twists of $\mathbb{Z}_p(1)$