Quadratic twists of genus one curves

For a given irreducible and monic polynomial $f\left(x\right)\in Z\left[x\right]$ of degree 4, we consider the quadratic twists by square-free integers q of the genus one quartic $H:y^2=f\left(x\right)$$H_q:q{y}^2=f\left(x\right).$

Let L denote the set of positive square-free integers q for which $H_q$ is everywhere locally solvable. For a real number x, let $L\left(x\right)=\#\{q\in L:q\le x\}$ be the number of elements in L that are less than or equal to x.

In this paper, we obtain that$L\left(x\right)=c_f\frac{x}{{(\ln x)}^m}+O\left(\frac{x}{{(\ln x)}^{\alpha }}\right)$ for some constants $c_f>0$, m and α only depending on f such that $m<\alpha \le 1+m$. We also express the Dirichlet series $F\left(s\right)={\sum }_{n\in L}{n}^{-s}$ associated to the set L in terms of Dedekind zeta functions of certain number fields.