Let
$$\displaystyle F(x,y,z) = \sum_{\substack{0 \leq i_1, i_2, i_3 \leq 2 \\ i_1 + i_2 + i_3 = 2}} a_{i_1, i_2, i_3} x_1^{i_1} x_2^{i_2} x_3^{i_3}$$
be a ternary quadratic form with integer coefficients. Let $Y$ denote the conic defined by $F = 0$ in $\mathbb{P}^2$, and for a given field $K$, let $Y(K)$ denote the set of $K$-points on $Y$. For each prime $p$, define the set $S_p$ as the union of the sets $\{(x_1, x_2, x_3) \in \mathbb{F}_p^3 : (x_1, x_2, x_3) \not \in Y(\mathbb{F}_p)\}$ and $\{(x_1, x_2, x_3) \in \mathbb{F}_p^3: (x_1, x_2, x_3) \in Y(\mathbb{F}_p), x_1 \text{ is a quadratic residue modulo } p\}$. For a positive parameter $Z$, define
$$\displaystyle \mathcal{S}_F(Z) = \{(x_1, x_2, x_3) \in \mathbb{Z}^3: \max\{|x_i|\} \leq Z, (x_1, x_2, x_3) \in S_p \text{ for all } p \}.$$
Is there an estimate for the count $N_F(Z) = \# \mathcal{S}_F(Z)$?
