= Solution
For negligible final <fermion> masses, the <chiral Z-boson partial width> is proportional to $g_L^2+g_R^2$. Both channels have one color state. An <Electron> has $Q=-1$, $T_{3L}=-1/2$ and $T_{3R}=0$, giving
$$
g_L^e=-\frac12+0.23=-0.27,\qquad g_R^e=0.23.
$$
For one active <neutrino> species, $Q=0$, $T_{3L}=1/2$ and there is no coupled right-handed neutrino field, so $g_L^\nu=1/2$, $g_R^\nu=0$. The common coupling normalization and <phase space> cancel in the ratio:
$$
\frac{\Gamma(Z\to\nu_e\bar\nu_e)}{\Gamma(Z\to e^-e^+)}=\frac{(1/2)^2}{(-0.27)^2+(0.23)^2}=1.9873.
$$
The same total <decay width> appears in the two <branching fractions>, so
$$
\boxed{\operatorname{Br}(Z\to\nu_e\bar\nu_e)=1.9873\times3.36\%=6.68\%.}
$$
There is no factor of three for a single specified flavour. Summing three active <neutrino> flavours would give an invisible <branching fraction> of approximately $20.0\%$ in this approximation.
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