Anisotropic theta functional equation (${\theta }_{\Lambda }(A)=\mathrm{covol}(\Lambda {)}^{-1}(\det A{)}^{-1/2}{\theta }_{{\Lambda }^{\ast }}(A^{-1})$)

For a full Euclidean lattice and a positive definite real symmetric matrix $A$, the Gaussian theta sum obeysUse the Fourier transform kernel $e^{-2\pi ix\cdot z}$, the identity $\widehat{e^{-\pi x^{T}Ax}}(z)=(\det A{)}^{-1/2}{e}^{-\pi z^T{A}^{-1}z}$, and the Poisson summation formula for a Euclidean lattice. No integrality or self-duality of the Euclidean lattice is assumed.