  This article provides some solutions to Chinburg's conjectures by studying a
sequence of the Mahler measure of multivariate exact polynomials. These
conjectures assert that for every odd quadratic Dirichlet Character of
conductor $f$, $\chi_{-f}=\left(\frac{-f}{.}\right)$, there exists a bivariate
polynomial (or a rational polynomial in the weak version) whose Mahler measure
is a rational multiple of $L'(\chi_{-f},-1)$. To obtain such solutions for the
conjectures we study a polynomial family denoted by $P_d(x,y)$, whose Mahler
measure has been fully studied in \cite{mehrabdollahei2021mahler}, and
\cite{BGMP}. We demonstrate that the Mahler measure of $P_d$ can be expressed
as a linear combination of Dirichlet $L$-functions, which has the potential to
generate solutions to the conjectures. Specifically, we prove that this family
provides solutions for conductors $f=3,4,8,15,20,24$. Notably, $P_d$
polynomials provide intriguing examples where the Mahler measures are linked to
$L'(\chi,-1)$ with $\chi$ being an odd non-real primitive Dirichlet character.
These examples inspired us to generalize Chinburg's conjectures from real
primitive odd Dirichlet characters to all primitive odd characters. For the
generalized version of Chinburg's conjecture, $P_d$ polynomials provide
solutions for conductors $5,7,9$.
