This paper is devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. The authors consider the case when the two zones are separated by a straight line \(\Sigma\) and the singular point of the unperturbed system is in \(\Sigma\). It is proved that the maximum number of limit cycles that can appear up to a seventh-order perturbation is three. This upper bound is reached. For systems which include a first-order perturbation it is proved that when the period function, defined in the period annulus of the center, is not monotone then it has at most one critical period.