In this paper, we show that the zeros of sampled-data systems resulting from rapid sampling of continuous-time systems preceded by a zero-order hold (ZOH) are the roots of the Euler--Frobenius polynomials. Using known properties of these polynomials, we prove two conjectures of Hagiwara and co-workers, the first of which concerns the simplicity, negative realness and interlacing properties of the sampling zeros of ZOH- and first-order hold (FOH-) sampled systems. To prove the second conjecture, we show that in the fast sampling limit, and as the continuous-time relative degree increases, the largest sampling zero for FOH-sampled systems approaches $1/e$, where $e$ is the base of the natural logarithm.