We present the X-ray properties of a sample of 17 radio sources observed with the Chandra X-ray Observatory as part of a project aimed at studying the X-ray emission from their radio jets. In this paper, we concentrate on the X-ray properties of the unresolved cores. The sample includes 16 quasars (11 core-dominated and 5 lobe-dominated) in the redshift range z=0.30--1.96, and one low-power radio-galaxy at z=0.064. No diffuse X-ray emission is present around the cores of the quasars, except for the nearby low-power galaxy that has diffuse emission on a scale and with a luminosity consistent with other FRIs. No high-amplitude, short-term variability is detected within the relatively short Chandra exposures. However, 1510-089 shows low-amplitude flux changes with a timescale of $sim$25 minutes. The X-ray spectra of the quasar cores are generally well described by a single power law model with Galactic absorption. However, in six quasars we find soft X-ray excess emission below 1.6 keV. Interestingly, we detect an Fe K-shell emission line, consistent with fluorescent Kalpha emission from cold Iron, in one lobe- and two core-dominated sources. The average X-ray photon index for the quasars in the sample is Gamma=1.66 and dispersion, sigma=0.23. The average spectral slope for our sample is flatter than the slope found for radio-quiet quasars and for radio-loud AGNs with larger jet orientations; this indicates that beaming affects the X-ray emission from the cores in our sample of quasars.
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