On the functional equations of Siegel Eisenstein series of an odd prime level <i>p</i>

Let p be an odd prime. In this paper, we write down the functional equations of Siegel Eisenstein series of degree n, level p with quadratic or trivial characters. First, we study the action of [Formula: see text]-operator in the space of Siegel Eisenstein series to get [Formula: see text])-eigenfunctions. Second, we show that the functional equations for such [Formula: see text]-eigenfunctions are quite simple and easy to write down by studying the constant term of the Fourier expansion, together with the functional equations of Eisenstein series due to Langlands. Consequently, we show that the matrix that represents functional equations of Siegel Eisenstein series are given by the products of the strict upper triangular matrices and the anti-diagonal matrix.