As known to us all, L-shaped co-prime array (LCA) is a recently introduced two-dimensional (2-D) sparse array structure, which is extended from linear co-prime array (CA). Such sparse array geometry can be used for 2-D parameters estimation with higher degrees-of-freedom (DOF). However, in the scenario where several narrowband transmissions spread over a wide spectrum, existing technique based on LCA with Nyquist sampling may encounter a bottleneck for both analog and digital processing. To alleviate the burden of high-rate Nyquist sampling, a method of joint wideband spectrum and direction-of-arrival (DOA) estimation with compressed sampling based on LCA, which is recognized as LCA-based modulated wideband converter (MWC), is presented in this work. First, the received signal along each antenna is mixed to basebands, low-pass filtered and down-sampled to get the compressed sampling data. Then by constructing the virtual received data of 2-D difference coarray, we estimate the wideband spectrum and DOA jointly using two recovery methods where the first is a joint ESPRIT method and the other is a joint CS method. Numerical simulations illustrate the validity of the proposed LCA based MWC system and show the superiority.