An amplitude analysis of the decay $Lambda_b^0to D^0 p pi^-$ is performed in the part of the phase space containing resonances in the $D^0 p$ channel. The study is based on a data sample corresponding to an integrated luminosity of 3.0 fb$^{-1}$ of $pp$ collisions recorded by the LHCb experiment. The spectrum of excited $Lambda_c^+$ states that decay into $D^0 p$ is studied. The masses, widths and quantum numbers of the $Lambda_c(2880)^+$ and $Lambda_c(2940)^+$ resonances are measured. The constraints on the spin and parity for the $Lambda_c(2940)^+$ state are obtained for the first time. A near-threshold enhancement in the $D^0 p$ amplitude is investigated and found to be consistent with a new resonance, denoted the $Lambda_c(2860)^+$, of spin $3/2$ and positive parity.
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