The cubic Pell equation L-function

Abstract

For $d>1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb{Q}\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s)>1$, and its coefficients vanish except at values corresponding to integral solutions of $m{x}^3-dn{y}^3=1$ in $\mathbb{Q}\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the meromorphic continuation of $L_d(s)$ to $\text{Re}(s)>\frac{1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s)\ll |s{|}^{\frac{7}{2}}$ and has a possible simple pole at $s=\frac{2}{3}$, possible poles at the zeros of a certain Appell hypergeometric function, with no other poles. We conjecture that the latter case does not occur, so that $L_d(s)$ has no other poles with $\text{Re}(s)>\frac{1}{2}$ besides the possible simple pole at $s=\frac{2}{3}$.