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Hi!

Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the global theta lift of trivial character of U(1) to U(3) vanish.

In the paper of Gelbart anf Rogalwski,"L-functions and Fourier-Jacobi coefficients for the unitary group U(3)"', propsition 3.4.1 says that it does nonvanish. If it does not vanish, I am also wondering whether it is cuspidal. Theorem 5.1 of that paper says that it is equivalent to $L(1/2,\gamma^3)$ where $\gamma$ is a character of $A_{E}^*$ and its restriction to $A_{F}^*$ is a character associated to the quadratic character of E/F by class field theory. Then this $L(1/2,\gamma^3)$ does vanish or non-vanish?

Since I am the beginner in this area, this question might be very silly, but I will be heartly thankful if anyone sheds me some light on this.

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