An exotic narrow state in the \(D^0D^0\pi^+\) mass spectrum just below the \(D^{*+}D^0\) mass threshold is studied using a data set corresponding to an integrated luminosity of 9 fb\(^{-1}\) acquired with the LHCb detector in proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV. The state is consistent with the ground isoscalar \(T^+_{cc}\) tetraquark with a quark content of \(cc\bar{u}\bar{d}\) and spin-parity quantum numbers \(\mathrm{J}^{\mathrm{P}}=1^+\). Study of the \(DD\) mass spectra disfavours interpretation of the resonance as the isovector state. The decay structure via intermediate off-shell \(D^{*+}\) mesons is confirmed by \(D^0\pi^+\) mass distribution. The mass of the resonance and its coupling to the \(D^{*+}D^0\) system are analysed. Resonance parameters, the pole position, scattering length, effective range and compositeness are measured to reveal important information about the nature of the \(T^+_{cc}\) state. In addition, an unexpected dependence of the production rate on track multiplicity is observed.