So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a polynomial of degree $1$). So the Hasse-Weil zeta function of $X_0$ is given by
$$
L_{X_0}(s):=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s).
$$
Now if one lets $\psi(z)=\sum_{n\in\mathbf{Z}}e^{i\pi n^2z}$ then $\psi(z)$ is a modular form of weight $1/2$ (over a suitable congruence group of $SL_2(\mathbf{Z})$) in the sense that
$$
(-iz)^{-1/2}\psi(-1/z)=\psi(z). \;\;\;\;(*)
$$
A straight forward computation which uses the definition of the Gamma function implies that
$$
\tilde{L}_{X_0}(s):=\int_{0}^{\infty}(\psi(it)-1)t^{s/2}\frac{dt}{t}=2\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
Using the functional equation $(*)$ one may deduce the meromorphic continuation and the functional equation of $\zeta(s)$ (invariance of $\tilde{L}_{X_0}(s)$ under $s\mapsto 1-s$).

Now let us take the scheme $X_1=\mathbf{P}^1$ over $Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_1$ at a prime $p$ is given by $Z_p(T)=(1-T)(1-pT)=1-\sigma_1(p)T+pT^2$ (a polynomial of degree $2$). It thus follows that the 
Hasse-Weil zeta function of $X_1$ is given by
$$
L_{X_1}(s)=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s)\zeta(s-1).
$$
Now let us look at the Eisenstein series of weight $2$ i.e.
$$
E_2(z)=(2\pi i)^{-2}\sum_{m,n}'\frac{1}{(mz+n)^2}:=\frac{-B_2}{2}+2\sum_{n\geq 1}\sigma_1(n)q_{z}^n,
$$
where $q_{z}=e^{2\pi iz}$. (Note that I don't get any convergence issue here since I take this $q$-expansion as the definition of $E_2(z)$). Note that $E_2(z)$ is "almost" a moldular form of weight $2$ (for the full congruence group $SL_2(\mathbf{Z})$) since 
$$
(-z)^{-2}E_2(-1/z)=E_2(z)-\frac{1}{2\pi iz}   \;\;\;\; (**)
$$
A straight forward computation similar to the one before implies that
$$
\tilde{L}_{X_1}(s):=\int_{0}^{\infty} (E_2(it)+B_2/2)t^{s}\frac{dt}{t}=2 (2\pi)^{-s}\Gamma(s)\zeta(s)\zeta(s-1).
$$
As before using $(**)$ one obtains the meromorphic continuation of $L_{X_1}(s)$ and its functional equation ($\tilde{L}_{X_1}(s)=-\tilde{L}_{X_1}(2-s)$, note the appearance of the sign $-1$). Note that this could already be deduced from what we know from $L_{X_0}(s)$.

Now there is no reason to stop here. So let $X_2=\mathbf{P}^2$ over $Spec(\mathbf{Z})$.
Then the Weil zeta function of $X_2$ at $p$ is $Z_p(T)=(1-T)(1-pT)(1-p^2T)$ (a polynomial of degree $3$). It thus follows that the 
Hasse-Weil zeta function of $X_2$ is given by
$$
L_{X_2}(s)=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s)\zeta(s-1)\zeta(s-2)=\sum_{n\geq 1}\frac{a_n}{n^s}
$$

Q1: Is it reasonable to expect the formal $q$-expansion $f(q_z)=\sum_{n\geq 1} a_n q_z^n$ to be related in some direct way to a modular form w.r.t. a suitable congruence subgroup of $GL_3(\mathbf{Z})$?

Q2: What about $X_n=\mathbf{P}^n$ in general?

**added**: Note that in the case of $X_0$ I'm really looking at $\tilde{\zeta}(s):=\zeta(2s)$ which is an $L$-function of weight $1/2$ in the sense that $\tilde{\zeta}(s)=\tilde{\zeta}(1/2-s)$ which is in accordance with the fact that $\psi(z)$ has weight $1/2$.