On the calculation of the ramified Siegel series

The ramified Siegel series is an important factor that appears in the Fourier coefficient of the Siegel Eisenstein series.Many formulas for the ramified Siegel series under various conditions are already known.However, an explicit formula for the general case has not yet been obtained.We derive a formula for the Siegel series with arbitrary dimension $n$, assuming that the additive character $\psi$ is primitive.Our results cover nonarchimedean, non-dyadic local fields $F$, including the case $F=\mathbb{Q}_p$.We also give explicit values of the ramified Siegel series for degrees $n=1, 2,$ and $3$.