The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over

I am just curious to know the origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation $f(s)=w(C/ℚ)f(2-s)=εf(2-s)$) of the curve $C$. I read several papers on this topic, but I cannot find where this root number come from. I wonder if this number has some relation with $a$ and $b$ in the equation of the curve $C$.