Lang–Trotter fixed-Frobenius-field conjecture for elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication, with conductor $N$, and let $F$ be an imaginary quadratic field. For primes of good reduction, define

$$S(E,F;x)=\#\{p\leq x:p\nmid N,\ F(E,p)=F\}.$$

Lang–Trotter fixed-Frobenius-field conjecture. There exists a constant $C(E,F)>0$ such that

$$S(E,F;x)\sim C(E,F)\frac{\sqrt{x}}{\operatorname{log} x}$$

as $x\to+\infty$. This predicts that each imaginary quadratic Frobenius field occurs with square-root-logarithmic frequency among primes of good reduction. The source attributes the conjecture to Lang and Trotter and cites detailed discussion and results concerning its formulation.