We show that, at the critical temperature, there is a class of Lee-Yang zeros of the partition function in a general scalar field theory, which location scales with the size of the system with a characteristic exponent expressed in terms of the isothermal critical exponent δ. In the thermodynamic limit the zeros belonging to this class condense to the critical point ζ=1 on the real axis in the complex fugacity plane while the complementary set of zeros (with Reζ<1) covers the unit circle. Although the aforementioned class degenerates to a single point for an infinite system, when the size is finite it contributes significantly to the partition function and reflects the self-similar structure (fractal geometry, scaling laws) of the critical system. This property opens up the perspective to formulate finite-size scaling theory in effective QCD, near the chiral critical point, in terms of the location of Lee-Yang zeros.