The superconducting state of the newly discovered superconductor K$_2$Cr$_3$As$_3$ with a quasi-one-dimensional crystal structure ($T_{bf c}sim$ 6 K) has been investigated by using magnetization and muon-spin relaxation or rotation ($mu$SR) measurements. Our analysis of the temperature dependence of the superfluid density obtained from the transverse field (TF) $mu$SR measurements fit very well to an isotropic $s$-wave character for the superconducting gap. Furthermore a similarly good fit can also be obtained using a $d$-wave model with line nodes. Our zero-field $mu$SR measurements do reveal very weak evidence of the spontaneous appearance of an internal magnetic field near the transition temperature, which might indicate that the superconducting state is not conventional. This observation suggests that the electrons are paired via unconventional channels such as spin fluctuations, as proposed on the basis of theoretical models of K$_2$Cr$_3$As$_3$. Furthermore, from our TF $mu$SR study the magnetic penetration depth $lambda_L$, superconducting carrier density $n_s$, and effective-mass enhancement $m^*$ have been estimated to be $lambda_L(0)$ = 454(4) nm, $n_s$ = 2.4$times$10$^{27}$ carriers/m$^3$, and $m^*$ = 1.75 $m_e$, respectively.
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