An enhancement of Zagier’s polylogarithm conjecture

Let $m\geq 2$ be a natural number and let $\mathcal {A}$ be an ideal class of an imaginary quadratic number field. Zagier and Gangl constructed $\mathbb {C}/\mathbb {Q}(m)$-valued invariants $I_{m}(\mathcal {A})$ which they named “the enhanced zeta value”, since the real part of $i^{m-1}I_{m}(\mathcal {A})$, after being multiplied by a certain elementary factor in terms of a factorial and a power of $2\pi$, equals the partial zeta value $\zeta (m,\mathcal {A})$. They also constructed the enhanced polylogarithm, a $\mathbb {C}/\mathbb {Q}(m)$-valued function on the $m$-th Bloch group $\mathcal {B}_{m}(\mathbb {C})$, and formulated an enhanced conjecture for $I_{m}(\mathcal {A})$ that gives a natural lift of the polylogarithm conjecture for $\zeta (m,\mathcal {A})$ to a conjectural equality in $\mathbb {C}/\mathbb {Q}(m)$. In this article, we define the Shintani L-function of two variables which is naturally regarded as a two-variable analog of the partial zeta function for imaginary quadratic fields. Then we study its analytic properties in order to construct $\mathbb {C}/\mathbb {Q}(1)$-valued invariants $\Lambda _{i}(1-m,\mathcal {A})$ ($i\in \{1,2\}$) for a ray class $\mathcal {A}$ using the first partial derivative of the Shintani L-function at $(1-m,1-m)$. From the construction, $\Lambda _{1}(1-m,\mathcal {A})$ and $\Lambda _{2}(1-m,\mathcal {A})$ are complex conjugate invariants that satisfy $\zeta ’(1-m,\mathcal {A})=\Lambda _{1}(1-m,\mathcal {A})+\Lambda _{2}(1-m,\mathcal {A})$. Then we prove the main theorem of this article about the equality between Zagier and Gangl’s enhanced zeta value $I_{m}(\mathcal {A})$ and $\Lambda _{1}(1-m,\mathcal {A})$, by explicit calculation of the Fourier expansion of the partial derivative of the Shintani L-function. Finally, we formulate the enhanced conjecture for the ray class invariants $\Lambda _{i}(1-m,\mathcal {A})$, by which we expand Zagier-Gangl’s original conjecture. We also give several numerical examples to verify the correctness of our enhanced conjecture.