Disclaimer: I first asked this question on Mathoverflow but I was told it was off-topic for that site, so I ask it here.
I try to understand correctly the notion of scheme, as Serre in the second volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem that says the zeta function $\zeta(X,s)$ of a scheme $X$ converges absolutely in the right half plane $\Re(s)\gt \dim X$, which is interesting for two reasons:
1) the Hasse-Weil zeta function of a rational elliptic curve is, up to normalization, an element of the Selberg class of degree two, and, as far as I know, before the modularity theorem was proved, such an L-function was only known to converge absolutely in the right half plane $\Re(s)>2$,
2) every element of the Selberg class converges absolutely in the right half-plane $\Re(s)\gt 1$, and thus in any right half plane of the form $\Re(s)\gt m$ where $m$ is any positive integer.
Hence my questions:
A) does the notion of fibered product of schemes imply that, if $X$ and $Y$ are two schemes of finite type, the dimension of the fibered product of $X$ and $Y$ is the product of the dimensions of $X$ and $Y$?
B) is any (local factor of an) element of the Selberg class the Zeta function of a scheme of finite type?
C) is the degree conjecture for the Selberg class motivated by the fact that the degree of an element thereof should be the dimension of a scheme?
Thanks in advance.