I have to plot this function:

Q[ω_] := ArcTan[1/ω]/(1 + 1/2*Log[(1 + ω^2)/ω^2])

And I did it through

Plot[Q[ω], {ω, 0, 3},
  PlotRange -> {0, 1}, 
  PlotStyle -> {Black}]

In a very simple way.

The output is the following:

But here is the problem: the function, as $\omega \to 0$ must be zero.

Why the plot doesn't show that behaviour?

The convergence to zero is really slow (to say: for $\omega \to 10^{-309}$ I just obtained $0.0022$). Is there a way to make the plot to start from zero?