I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim N(\mu _y,\sigma_y)$$ where $\mu _x\neq 0, \sigma _x\neq 1, \mu _y\neq 0,\sigma _y\neq 1$
For the standard case you can find the solution here , but I could not find any solution for the non-standard one. I would appreciate any help. Thank you.