I think the usual proof for the asymptotic number of zeros of the Riemann zeta function
$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\rho) \ \in\ (-1,2)\end{array}\right\} = \frac{ T\ln T}{2\pi}-\frac{1+\log 2\pi}{2\pi}T+\mathcal{O}(\ln T)$$
works for any Dirichlet series in the **extended Selberg class $S^{\#}$** (for the Selberg class it is a well-known result), that is Dirichlet series having a functional equation with gamma factors, and being analytic except possibly a pole at $s=1$. More precisely :

$F(s) = \sum_{n=1}^\infty a_n n^{-s}$ converges absolutely on $Re(s)>1$, and $F(s) (s-1)^m$ is entire of finite order,

with $\gamma(s) = Q^s\prod_{j=1}^{k} \Gamma(\omega_j s+\mu_j), \omega_j > 0$ and $\Phi(s) = \gamma(s)F(s) : \quad \Phi(s) = \xi\, \overline{\Phi(1-\overline{s})}$

Then $$N_F(T) = \frac{d_F}{2\pi} T \ln T+\frac{c_F}{2\pi} T+\mathcal{O}(\ln T), \qquad d_F=\sum_{j=1}^m \omega_j, \quad c_F = \ln |Q|^2-1-\ln 2 d_F$$ where $N_F(T) = \#\left\{\rho : \ F(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\rho) \ \in\ (-\delta_F,1+\delta_F)\end{array}\right\}$ is the number of zeros, and $ \delta_F$ is chosen such that $\sum_{n=2}^\infty |a_n| n^{-1-\delta_F} < |a_1|$ i.e. $\text{arg}(F(1+\delta_F+it) = \mathcal{O}(1)$

**Questions :**

Do you have a reference confirming this ? And if I missed something, what additional hypothesis on $F(s)$ are needed ?

That the asymptotic number of zeros depends on the functional equation not on the Euler product, what does it tell us, about $\zeta(s)$ and the L-functions ?

*The asymptotics for the zeros allow us to write $\frac{F'}{F}(s) = \sum_{|Im(\rho)-t)| < A} \frac{1}{s-\rho}+\mathcal{O}(\ln t)$ in the critical strip, and then to look at how different constraints (Euler product, growth rate estimates for $F,F',1/F$) interact with those density of zeros. Assuming the GRH, we are probably also allowed to make some general statements about the number of zero crossings of $\Phi(1/2+it)$, and to link it to some properties of modular forms on $Re(\tau) =0$.*What happens if I add to $S^{\#}$ the constraint that there is some $l$ such that $\frac{1}{\zeta(\sigma)^l} < |F(s)| \le \zeta(\sigma)^l$ for every $Re(s)=\sigma > 1$ ?

(

*it could mean that $F(s)$ has an Euler product of the form $\prod_{j=m}^l \prod_p (1-\alpha_{j}(p)p^{-s})^{-1}$*where $|\alpha_j(p)|\le 1$)