Suppose $\chi$ is a real Drichlet character of modulus $N$, \begin{equation} \chi: (\mathbb{Z}/N\mathbb{Z})^* \rightarrow \mathbb{C} \end{equation} which induces an Artin representation \begin{equation} \rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}^1(\mathbb{C}) \end{equation} Since $\chi$'s value is 1 or -1, both of which are in $\mathbb{Q}_{\ell}$, so there is a representation defined similarly as $\rho$, \begin{equation} \rho_{\ell}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}^1(\mathbb{Q}_{\ell}) \end{equation} Is the characteristic polynomial of the geometric Frobenius at $p$ (for almost all $\ell$) of the form? \begin{equation} 1-\chi(p)\, T \end{equation} which is just 1 for some $p$, a polynomial of degree 0.

Is there a (mixed) motive, whose $\ell$-adic realisation is the representation $\rho_{\ell}$?