We present IRAM 30m sensitive upper limits on CO emission in the ram pressure stripped dwarf Virgo galaxy IC3418 and in a few positions covering HII regions in its prominent 17 kpc UV/Ha gas-stripped tail. In the central few arcseconds of the galaxy, we report a possible marginal detection of about 1x10^6 M_sun of molecular gas (assuming a Galactic CO-to-H_2 conversion factor) that could correspond to a surviving nuclear gas reservoir. We estimate that there is less molecular gas in the main body of IC3418, by at least a factor of 20, than would be expected from the pre-quenching UV-based star formation rate assuming the typical gas depletion timescale of 2 Gyr. Given the lack of star formation in the main body, we think the H_2-deficiency is real, although some of it may also arise from a higher CO-to-H_2 factor typical in low-metallicity, low-mass galaxies. The presence of HII regions in the tail of IC3418 suggests that there must be some dense gas; however, only upper limits of < 1x10^6 M_sun were found in the three observed points in the outer tail. This yields an upper limit on the molecular gas content of the whole tail < 1x10^7 M_sun, which is an amount similar to the estimates from the observed star formation rate over the tail. We also present strong upper limits on the X-ray emission of the stripped gas in IC3418 from a new Chandra observation. The measured X-ray luminosity of the IC3418 tail is about 280 times lower than that of ESO 137-001, a spiral galaxy in a more distant cluster with a prominent ram pressure stripped tail. Non-detection of any diffuse X-ray emission in the IC3418 tail may be due to a low gas content in the tail associated with its advanced evolutionary state and/or due to a rather low thermal pressure of the surrounding intra-cluster medium.
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