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In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of <i>℘</i>, where <i>℘</i> is the Weierstrass <i>℘</i>-function attached to a rank two lattice of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>, yield rational equivariant functions. Our concern in this survey is to provide certain examples of rational equivariant functions. In this sense, we establish a criterion in order to determine the rationality of equivariant functions derived from ratios of modular functions of low weight. Modular forms play an important role in number theory and many areas of mathematics and physics.