  We attach a certain $n \times n$ matrix $A_n$ to the Dirichlet series
$L(s)=\sum_{k=1}^{\infty}a_k k^{-s}$. We study the determinant, characteristic
polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of
$A_n$ can be understood as a weighted sum of the first $n$ coefficients of the
Dirichlet series $L(s)^{-1}$. We give an interpretation of the partial sum of a
Dirichlet series as a product of eigenvalues. In a special case, the
determinant of $A_n$ is the sum of the M\"obius function. We disprove a
conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.
