Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $GL_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$ View $\mathscr{B}(\chi, \chi^{-1})$ in its Whittaker model $W(\pi, \psi)$ for a fixed additive character $\psi$ on $F$. Let $W\begin{pmatrix} y & \\ & 1 \end{pmatrix}$ be a vector in the Whittaker model. Then I want an explicit description of $$(\pi(w)W)\begin{pmatrix} y & \\ & 1 \end{pmatrix},$$ in either the Whittaker or Kirillov model of $\pi$. From trying to find an answer to this question I have learned that I may need to utilize a sort of $\GL_2 \times \GL_1$ local functional equation which I think would take the shape $$\gamma(\pi \otimes \chi, 1/2, \psi)\int_{F^\times} W(t)\chi(t)\,d^\times t = \int_{F^\times} (\pi(w)W)(t)\omega^{-1}(t)\chi^{-1}(t)\,d^\times t,$$ now viewing $W$ in its Kirillov model with $\omega$ the central character. Assuming for the moment the central character is trivial, Mellin inversion should yield $$(\pi(w)W)(y) = \int_{\chi} \chi(y)\gamma(\pi \otimes \chi, 1/2, \psi)\left(\int_{F^\times} W(t)\chi(t)\,d^\times t\right)\,d\chi.$$ This leads me to ask two questions:

1. How to explicitly integrate over characters $\chi$? What does this integration mean?
2. What is $\gamma(\pi \otimes \chi, 1/2, \psi)$ explicitly and how can I calculate it in a way that is amenable to then integrating with it?

I would appreciate either direct answers or hints or being pointed to a reference that would answer these questions.