The first moment of primitive cubic L $L$ upper L-functions with fixed genus

Source abstract

Abstract We study the first moment of primitive cubic L $L$ upper L -functions over F q ( T ) $\mathbb {F}_q(T)$ double struck upper F Subscript q Baseline left parenthesis upper T right parenthesis in the non-Kummer setting q ≡ 2 mod 3 $q\equiv 2\ {\mathrm{mod}}\ 3$ q identical to 2 mod 3 . More precisely, we study the sum ∑ χ primitive cubic genus ( χ ) = g L q ( 1 2 , χ ) , $$\begin{align*} \sum_{\substack{\chi\ \mathrm{primitive}\ \mathrm{cubic}\\ \mathrm{genus}(\chi)=g}}L_q\left(\frac{1}{2}, \chi\right), \end{align*}$$ where L q ( s , χ ) $L_q(s,\chi )$ upper L Subscript q Baseline left parenthesis s comma chi right parenthesis denotes the L $L$ upper L -function associated with primitive cubic character χ $\chi$ chi . Using a double Dirichlet series, we obtain an asymptotic formula with an error term of size q ( 7 8 + ε ) g $q^{(\frac {7}{8}+\varepsilon )g}$ q Superscript left parenthesis seven eighths plus epsilon right parenthesis g . The argument avoids an explicit residue computation and clarifies why the poles at u 3 = 1 $u^3=1$ u cubed equals 1 do not contribute, thereby resolving the question left in Remark 4.6 of David et al. (2019, algebra number theory ).