I am trying to show that a normal distribution with parameters $\mu = 0$ and variance $\theta$ is not complete. I am looking for a function $u(x)$ that is not equal to 0 such that $\mathbb E(u(x)) = 0$.
I have done some research on this problem and I have found that $\bar{X}$ and $S$ (sample standard deviation) are independent and can help yield me a function that will give me $\mathbb E(u(x)) = 0$. I know that $\bar{X}$ is normally distributed with mean 0 and variance $\theta/n$. I know that $S^2$ has a chi-squared distribution. I was trying to take an expectation $\mathbb E(\bar{X} S^2)$ to yield a value of 0, but I am not sure how exactly to do that.
Also, would this approach be correct in showing that it is not complete for $\theta$?
EDIT: An iid sample is taken and theta > 0.