Let $G$ be a finite abelian group and let $\varnothing \neq A \subset \mathbb Z$. The $A$-weighted Davenport constant of $G$ is the smallest positive integer ${\sf D}_A(G)$ such that every sequence $x_1 \boldsymbol{\cdot} {\dots} \boldsymbol{\cdot} x_{{\sf D}_A(G)}$ over $G$ has a non-empty subsequence $(x_{j_i})_i$ such that ${\varepsilon_1} x_{j_1} + {\varepsilon_2} x_{j_2} + {\dots} + {\varepsilon_t} x_{j_t} = 0$ for some $\varepsilon_1, \varepsilon_2, {\dots}, \varepsilon_t \in A$. In this paper, we obtain both upper and lower bounds for ${\sf D}_{\{1,s\}}(C_n^k)$, where $C_n$ denotes the cyclic group of order $n$, $s^2 \equiv 1 \pmod n$ and $s \not\equiv \pm1 \pmod n$. These bounds become sharp in some "small" cases.