Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves

For a finite field ${\mathbb{F}}_q$ of characteristic $p\ge 5$ and $K={\mathbb{F}}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy-t^{d}y=x^3$ for all integers $d$ coprime to $q$. We provide an explicit expression for the $L$-functions of these curves in terms of Jacobi sums. Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer-Siegel theorem. More precisely, we estimate the asymptotic growth of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) by its N'eron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to \mathrm{\infty }$.