In this work, we study the null controllability by one-sided boundary controls of one-dimensional hyperbolic balance laws with constant coefficients. Our first result shows that, when the system has only one negative or positive speed, the minimal null control time of such systems depends on some orthogonality conditions for a particular sequence. This sequence is explicit in function of the coefficients of the system but it is defined by a nonlinear recurrence relation. Our second result then completes the previous one by giving explicit bounds on the number of orthogonality conditions that have to be checked in two nontrivial situations. The proofs rely on a careful analysis of the so-called kernel equations associated with the system, including a new well-posedness result. Our results are also valid for the finite-time stabilization property.