In the book "Algebraic Geometry and Arithmetic Curves," Liu mentions in passing that the minimal Weierstrass model of an elliptic curve plays a similar role to the canonical model of curve of (arithmetic) genus $\geq2$. As far as I can tell the meaning of this comment is their respective roles in desingularization, and my question is how much further the comparison can be drawn?

For example, one defines the Hasse-Weil zeta function as the product of zeta functions of the local minimal Weierstrass equations and, if the elliptic curve has a global minimal Weierstrass equation, this agrees with the (Hasse) zeta function when the model is viewed as a scheme of finite type over $\mathbb{Z}$. This model is related to all others through birational geometry, and so the zeta functions of all models differ by a product of finitely many zeta functions of affine lines. As the Hasse-Weil zeta function is connected to the L-function of the curve through the equation $\zeta_E(s)=\frac{\zeta_k(s)\zeta_{k}(s-1)}{L_E(s)}$, where $k$ is the base field, the passage just described links the L-function to the zeta function of any regular model.

Is a similar passage known to exist for curves $C$ of arbitrary genus? Taking the L-function to be defined as that associated to $H^1$ easily gives us the expression above for the ``Hasse-Weil'' zeta function of the curve, which I take to be defined as the alternating product of the L-functions of cohomology groups (maybe there is a better definition?), the question is, does the zeta function of the canonical model agree with this Hasse-Weil zeta?