Chinburg's weak and strong conjectures for Mahler measures

Let $-f<0$ be a fundamental discriminant, and let $\chi_{-f}$ be the associated odd, primitive, quadratic Dirichlet character of conductor $f$, with $L$-function $L(\chi_{-f},s)$. For a rational function $P\in\mathbf{C}(x,y)^\times$, let its logarithmic Mahler measure be

$$\operatorname{m}(P)=\frac{1}{(2\pi\mathrm{i})^2}\int_{\mathbf{T}^2}\log|P(x,y)|\frac{\mathrm{d}x}{x}\frac{\mathrm{d}y}{y}.$$

Chinburg's conjectures. For every fundamental discriminant $-f<0$, there exists a bivariate rational function $P\in\mathbf{Z}(x,y)$ and a rational number $r\in\mathbf{Q}^{\times}$ such that

$$\operatorname{m}(P)=rL'(\chi_{-f},-1).$$

In the strong form, one may take $P\in\mathbf{Z}[x,y]$. These conjectures relate Mahler measures to special values of Dirichlet $L$-functions; the strong form was known for only finitely many conductors before the examples studied in this paper, while the weak form is proved here if cyclotomic coefficients are allowed.