Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function.

Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its pure numerical motive $hX_{\mathbb{C}}$ (living in $\mathcal{M}_{num}(\mathbb{C})$). I am given to believe that there is a way to define zeta functions on pure numerical motives (although I can't say I understand the construction).

Are these two the same? Could this be? It seems like one would lose a lot of (arithmetic) information when base changing to $\mathbb{C}$. For example, doesn't this imply that one would get the same function if one were to take the Hasse-Weil zeta function for any other $\mathbb{Q}$-model of $X_{\mathbb{C}}$?

Am I misunderstanding, or is this truly an uncanny phenomena? Is the construction conjectural? If not -- then it must mean that $X_{\mathbb{C}}$, a geometric object, contains all of the arithmetic information of a $\mathbb{Q}$-model of it. Why should that be true?