The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem

Our main objective in this paper is to study the average rank of the $2$-Selmer group of the elliptic curve associated with the $\frac{\pi}{3}$-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed $2$-Selmer groups, which has several consequences towards the average of $2$-Selmer ranks and $\frac{\pi}{3}$-congruent number problem. Indeed, we could find an unconditional positive density of $2$-Selmer rank being $1$ or $3$, among the positive square-free integers $n\equiv 13\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$ and an unconditional positive density of $2$-Selmer rank being $0$ or $2$, among the positive square-free integers $n\equiv 5\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$.