  We present a detailed investigation of the momentum-dependent self-energy
Sigma(k) at zero frequency of weakly interacting bosons at the critical
temperature T_c 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 << k_c and the short-wavelength regime k >> k_c, where k_c is the
crossover scale. In the critical regime, the self-energy correctly approaches
the asymptotic behavior Sigma(k) \propto k^{2 - eta}, and in the
short-wavelength regime the behavior is Sigma(k) \propto k^{2(D-3)} in D>3. In
D=3, we recover the logarithmic divergence Sigma(k) \propto ln(k/k_c)
encountered in perturbation theory. Our approach yields the crossover scale k_c
as well as a reasonable estimate for the critical exponent eta in D=3. From our
scaling function we find for the interaction-induced shift in T_c in three
dimensions, Delta T_c / T_c = 1.23 a n^{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 eta \approx 0.0513. We further calculate the constant lim_{k->0}
Sigma(k)/k^{2-eta} and find good agreement with recent Monte-Carlo data.
