We present a detailed investigation of the momentum-dependent self-energy Σ(k) at zero frequency of weakly interacting bosons at the critical temperature Tc of Bose-Einstein condensation in dimensions 3 ≤ D < 4. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k ≪ kc and the short-wavelength regime k ≫ kc, where kc is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior Σ(k) ∝ k 2−η , and in the short-wavelength regime the behavior is Σ(k) ∝ k 2(D−3) in D > 3. In D = 3, we recover the logarithmic divergence Σ(k) ∝ ln(k/kc) encountered in perturbation theory. Our approach yields the crossover scale kc as well as a reasonable estimate for the critical exponent η in D = 3. From our scaling function we find for the interaction-induced shift in Tc in three dimensions, ∆Tc/Tc = 1.23an 1/3 , where a is the s-wave scattering length and n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D = 3 and extend our truncation scheme of the renormalization group equations by including the six-and eight-point vertex, which yields an improved estimate for the anomalous dimension η ≈ 0.0513. We further calculate the constant lim k→0 Σ(k)/k 2−η and find good agreement with recent Monte-Carlo data.