Let $X = (X_1,X_2,\dots, X_n)$ be a sample from $N(\alpha\sigma, \sigma^2)$, where $\alpha$ is known real number. Show that the family of distributions of $T(X) = (\sum X_i, \sum X_i^2)$ is not complete for $\sigma$.
I am trying to show some counterexample for $g(X)$ such that $E(g(T))=0$ but $g$ is not $0$ a.e., but could not find any. Thanks for any help.