Superconductivity in noncentrosymmetric compounds has attracted sustained interest in the last decades. Here we present a detailed study on the transport, thermodynamic properties and the band structure of the noncentrosymmetric superconductor La$_7$Ir$_3$ ($T_c$ $sim$2.3 K) that was recently proposed to break the time-reversal symmetry. It is found that La$_7$Ir$_3$ displays a moderately large electronic heat capacity (Sommerfeld coefficient $gamma_n$ $sim$ 53.1 mJ/mol $text{K}^2$) and a significantly enhanced Kadowaki-Woods ratio (KWR $sim$ 32 $muOmega$ cm mol$^2$ K$^2$ J$^{-2}$) that is greater than the typical value ($sim$ 10 $muOmega$ cm mol$^2$ K$^2$ J$^{-2}$) for strongly correlated electron systems. The upper critical field $H_{c2}$ was seen to be nicely described by the single-band Werthamer-Helfand-Hohenberg model down to very low temperatures. The hydrostatic pressure effects on the superconductivity were also investigated. The heat capacity below $T_c$ reveals a dominant s-wave gap with the magnitude close to the BCS value. The first-principles calculations yield the electron-phonon coupling constant $lambda$ = 0.81 and the logarithmically averaged frequency $omega_{ln}$ = 78.5 K, resulting in a theoretical $T_c$ = 2.5 K, close to the experimental value. Our calculations suggest that the enhanced electronic heat capacity is more likely due to electron-phonon coupling, rather than the electron-electron correlation effects. Collectively, these results place severe constraints on any theory of exotic superconductivity in this system.
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