Zeros of Eisenstein series for the Fricke group $\Gamma_0^{*}(5)$

Let $k \geqslant 4$ be an even integer. For the Eisenstein series $E_{k,5}^{*}(z)$ associated with the Fricke group $\Gamma_0^{*}(5)$, let $\mathbb{F}^{*}(5)$ be the fundamental domain and let

$$A_5^{*}:=\mathbb{F}^{*}(5)\cap\{z\in\mathbb{C}:|z|=1/\sqrt{5}\text{ or }|z|=1/(2\sqrt{5})\}.$$

Zeros-on-the-arc conjecture for $\Gamma_0^{*}(5)$. All of the zeros of $E_{k,5}^{*}(z)$ in $\mathbb{F}^{*}(5)$ lie on the arc $A_5^{*}$. This concerns the extension of the Rankin–Swinnerton-Dyer phenomenon from the full modular group to Fricke groups; the paper presents it as a conjecture for $p=5$, while the analogous result for $p=2,3$ had already been proved.