Given i.i.d. samples $X_1,...,X_n$ from Normal$(\sigma,1)$, find the UMVUE of $g(\sigma)=P_\sigma(Z\leq 0)$.
I tried to use Lehman-Scheffe theorem. We now that $\sum_1^nX_i$ is sufficient and complete for $\sigma$, and we could take $T(X)=1_{\{X_1<0\}}$ as an unbiased estimator for $P(Z\leq 0)$. Then by this theorem, $S(X)=E[T(X)|\sum_1^nX_i]$ is the unique UMVUE for $g(\sigma)$. The point I got stuck is calculating the conditional expectation $E[1_{\{X_1<0\}}|\sum_1^nX_i]$.
Thanks in advance for any help/hint/suggestion.