  We report on the pressure-induced unconventional superconductivity in the
heavy-fermion antiferromagnet CeIn3 by means of nuclear-quadrupole-resonance
(NQR) studies conducted under a high pressure. The temperature and pressure
dependences of the NQR spectra have revealed a first-order quantum-phase
transition (QPT) from an AFM to PM at a critical pressure Pc=2.46 GPa. Despite
the lack of an AFM quantum critical point in the P-T phase diagram, we
highlight the fact that the unconventional SC occurs in both phases of the AFM
and PM. The nuclear spin-lattice relaxation rate 1/T1 in the AFM phase have
provided evidence for the uniformly coexisting AFM+SC phase. In the HF-PM phase
where AFM fluctuations are not developed, 1/T1 decreases without the coherence
peak just below Tc, followed by a power-law like T dependence that indicates an
unconventional SC with a line-node gap. Remarkably, Tc has a peak around Pc in
the HF-PM phase as well as in the AFM phase. In other words, an SC dome exists
with a maximum value of Tc = 230 mK around Pc, indicating that the origin of
the pressure-induced HF SC in CeIn3 is not relevant to AFM spin fluctuations
but to the emergence of the first-order QPT in CeIn3. When the AFM critical
temperature is suppressed at the termination point of the first-order QPT, Pc =
2.46 GPa, the diverging AFM spin-density fluctuations emerge at the critical
point from the AFM to PM. The results with CeIn3 leading to a new type of
quantum criticality deserve further theoretical investigations.
