Abstract This article provides partial solutions to Chinburg’s conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f, -₅= (-f. ) χ - f = - f. , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of L' (-₅, -1) L ′ (χ - f, - 1). We prove that the Mahler measure of a polynomial family, denoted by Pd P d, can be expressed as a linear combination of the derivatives of Dirichlet L -functions. Specifically, this family provides solutions to the conjectures for conductors f=3, 4, 8, 15, 20 f = 3, 4, 8, 15, 20, and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials Pd P d provide solutions for conductors 5, 7, and 9.
Bertin et al. (Sun,) studied this question.