What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each with its own gamma factor)

**In the end, this question is about understanding the BSD conjecture for abelian varieties over $\mathbf{Q}$ a bit better.**

If $X$ is an abelian variety over $\mathbf{Q}$, is there a Birch and Swinnerton-Dyer conjecture for the **zeta function** of its Néron model? Is it related to the Birch and Swinnerton-Dyer conjecture for the Hasse-Weil $L$-function of $X$?

**EXAMPLE:** let $E$ be an elliptic curve over $\mathbf{Q}$, $\mathcal{E}$ its Néron model over $\text{Spec}(\mathbf{Z})$.

The **zeta function** of $\mathcal{E}$ is, by definition:
$$Z(\mathcal{E}/\mathbf{Z},s) := \prod_{(p)\in\text{Spec}(\mathbf{Z})^0}Z(\mathcal{E}_p,p^{-s}),$$
where $\text{Spec}(\mathbf{Z})^0$ is the set of closed points of $\text{Spec}(\mathbf{Z})$, $\mathcal{E}_p$ is the mod $p$ fiber of $\mathcal{E}$, a variety over $\mathbf{F}_p$, and

$$Z(\mathcal{E}_p, T) = \frac{1 - a_pT + pT^2}{(1-T)(1-pT)},$$
while for **the Hasse-Weil $L$-function**
$$L(E/\mathbf{Q},s) := \prod_{p\ {\rm good}}\frac{1}{1-a_pp^{-s} + p^{1-2s}}\cdot\prod_{p\ {\rm split\ mult.}}\frac{1}{1-p^{-s}}\cdot\prod_{p\ {\rm non-split\ mult.}}\frac{1}{1+p^{-s}}\cdot\prod_{p\ {\rm additive}}1.$$
where $a_p$ is the size of $\mathcal{E}_p(\mathbf{F}_p)$.

It seems: $$(*)\ \ \ Z(\mathcal{E}/\mathbf{Z},s) = \frac{\zeta(s)\zeta(s-1)}{L(E/\mathbf{Q},s)}.$$

We can introduce gamma factors and get

$$\hat{Z}(\mathcal{E}/\mathbf{Z},s) = \frac{\hat{\zeta}(s)\hat{\zeta}(s-1)}{\hat{L}(E/\mathbf{Q},s)}$$ where $\hat{\zeta}(s)$ is the completed Riemann zeta function, and $\hat{L}(E/\mathbf{Q},s)$ is the Hasse-Weil $L$-function of $E$ multiplied by its gamma factor as described by Serre.

**PRECISE QUESTIONS:**

**(1.1):** Is the BSD conjecture about $\hat{L}(E/\mathbf{Q},s)$?
Ie. Is it true that $\text{ord}_{s = 1}\hat{L}(E/\mathbf{Q},s) = \text{ord}_{s=1}L(E/\mathbf{Q},s)$? (It looks the answer is yes because the gamma factor doesn't vanish at $s=1$). Is the BSD formula for $L$ and $\hat{L}$ the same? (ie. do we have $\text{Res}_{s=1}L(E/\mathbf{Q},s) = \text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s)$?)

**(1.2):** As a baby example of **(1.1)**. For $\zeta_K(s)$ the Dedekind zeta function of a number field $K$, and $\hat{\zeta}_K(s)$ the completed Dedekind zeta function, $\text{Res}_{s=1}\zeta_K(s)$ is predicted by the class number formula. What's the relation with $\text{Res}_{s=1}\hat{\zeta}_K(s)$? The former is
$$\text{Res}_{s=1}\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2} h_K R_K}{\omega_K\sqrt{|\Delta_K|}},$$
with usual meanings.

If we call $$\zeta_{K,\infty}(s) := |\Delta_K|^{s/2}\Gamma_{\mathbf{R}}(s)^{r_1}\Gamma_{\mathbf{C}}(s)^{r_2}$$ then the latter is $$\text{Res}_{s=1}\hat{\zeta}_K(s) = \zeta_{K,\infty}(1)\cdot\text{Res}_{s=1}\zeta_K(s) = 2^{r_1-r_2}\pi^{-r_1/2}\Gamma(1/2)^{r_1}\Gamma(1)^{r_2}\cdot \frac{ h_K R_K}{\omega_K}.$$

Does this expression mean anything? I was expecting to get a cleaner expression for the completed zeta function.

It looks the answer to **(1.1)**, then, should be
$$\text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s) = L_{\infty}(1)\cdot \text{Res}_{s=1}L(E/\mathbf{Q},s).$$

**(2): (the question I am most interested in)**

Is there a BSD conjecture for $\hat{Z}(\mathcal{E}/\mathbf{Z},s)$ (call it BSD$(\hat{Z})$)? Is it related to the BSD conjecture for $\hat{L}(E/\mathbf{Q},s)$ (call it BSD$(L)$)?

(specifically, do we have BSD$(\hat{Z})\Rightarrow$ BSD$(L)$?)

**(3):** What is a formula analogous to $(*)$, for higher dimensional abelian varieties and their zeta vs Hasse-Weil $L$-functions?

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**EDIT: an equivalent question I would be very interested to know the answer of.**

In the function field case and for elliptic curves, Tate proved the BSD conjecture (for each elliptic curve over a function field) is equivalent to the Artin-Tate conjecture for a canonical elliptic surface (over a finite field). I think this has been generalized to abelian varieties. Is it possible to prove something analogous over number fields? ie. is it possible to deduce the BSD conjecture for the Hasse-Weil $L$-function of an abelian variety from a number field analog of the Artin-Tate conjecture (where probably the leading coefficient formula involves torsion and volumes on higher Chow groups) for the **zeta** function of an appropriate arithmetic scheme?

(without making use of Weil-êtale cohomologies, etc)