Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multi-variable Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper we remove this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multi-variable Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then compared to the Euler characteristic of the combinatorial version of link Floer homology.