We consider in this article, a lopsided quintic polynomial vector field $X = -x {\partial\over \partial y} + y {\partial\over \partial x} + \sum_{i+j=5}(a_{ij} x^i y^j {\partial\over \partial x} ).$ We first compute the first non-zero derivative of the return map $r\mapsto L(r,\varepsilon)$. We study then the necessary and sufficient conditions for the existence of a center.