C-cyanomethanimine (HNCHCN), existing in the two $Z$ and $E$ isomeric forms, is a key prebiotic molecule, but, so far, only the $E$ isomer has been detected toward the massive star-forming region. Sagittarius B2(N) using transitions in the radio wavelength domain. With the aim of detecting HNCHCN in Sun-like-star forming regions, the laboratory investigation of its rotational spectrum has been extended to the millimeter-/submillimeter-wave (mm-/submm-) spectral window in which several unbiased spectral surveys have been already carried out. High-resolution laboratory measurements of the rotational spectrum of C-cyanomethanimine were carried out in the 100-420 GHz range using a frequency-modulation absorption spectrometer. We then searched for the C-cyanomethanimine spectral features in the mm-wave range using the high-sensitivity and unbiased spectral surveys obtained with the IRAM 30-m antenna in the ASAI context, the earliest stages of star formation from starless to evolved Class I objects being sampled. For both the $Z$ and $E$ isomers, the spectroscopic work has led to an improved and extended knowledge of the spectroscopic parameters, thus providing accurate predictions of the rotational signatures up to $sim$700 GHz. So far, no C-cyanomethanimine emission has been detected toward the ASAI targets, and upper limits of the column density of $sim$ 10$^{11}$--10$^{12}$ cm$^{-2}$ could only be derived. Consequently, the C-cyanomethanimine abundances have to be less than a few 10$^{-10}$ for starless and hot-corinos. A less stringent constraint, $leq$ 10$^{-9}$, is obtained for shocks sites. The combination of the upper limits of the abundances of C-cyanomethanimine together with accurate laboratory frequencies up to $sim$ 700 GHz poses the basis for future higher sensitivity searches around Sun-like-star forming regions.
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