NOVELTY - A lithium manganese complex oxide (1) has magnetized curve which is changed to -5.5T and measured at temperature of 5K after setting magnetic field (H) to 1-5.5T. When the magnetic field (H) is set to 5.5T, the magnetization rate of complex oxide per mol of manganese is maximum magnetization rate (Mmax), and when the magnetic field (H) is set to 0T, the magnetization rate of complex oxide per mol of manganese is residual-magnetization rate (Mr). The ratio (Mr/Mmaxx 100) of residual-magnetization rate (Mr) with respect to maximum magnetization rate (Mmax) is 2% or less.    USE - Lithium manganese complex oxide is used for lithium ion secondary battery (claimed) used as power supply for mobile telephone, personal computer, and hybrid vehicle.    ADVANTAGE - The lithium manganese complex oxide provides lithium ion secondary battery has high capacitance and charging/discharging cyclic characteristics.    DETAILED DESCRIPTION - A lithium manganese complex oxide has magnetized curve which is changed to -5.5T and measured at temperature of 5K after setting the magnetic field (H) to 1-5.5T. When the magnetic field (H) is set to 5.5T, the magnetization rate of lithium manganese complex oxide per mol of manganese is maximum magnetization rate (Mmax) (emu/mol-Mn), and when the magnetic field (H) is set to 0T, the magnetization rate of lithium manganese complex oxide per mol of manganese is residual-magnetization rate (Mr) (emu/mol-Mn). The ratio (Mr/Mmaxx 100) of residual-magnetization rate (Mr) with respect to maximum magnetization rate (Mmax) is 2% or less. The lithium manganese complex oxide has composition of formula: Li1+x-yMey+zMn2-x-zO4-n (1).    x=0-1/3;    y+z=0-1;    n=0-0.1;and    Me=aluminum, magnesium, iron, cobalt, nickel, copper, chromium or zinc.    INDEPENDENT CLAIMS are included for the following:    (1) manufacture of the lithium manganese complex oxide, which involves baking inorganic material used as lithium manganese complex oxide with predetermined calcination temperature, to obtain complex oxide for transmitting electrochemically inactive crystal phase to electrochemically active crystal phase, cooling at predetermined slow cooling rate, and oxidizing the complex oxide with predetermined heating temperature lower than the calcination temperature; and    (2) lithium ion secondary battery, which is provided with positive electrode having positive electrode active material containing lithium manganese complex oxide, negative electrode having negative electrode active material, and ionic-conduction medium which interposes between the positive electrode and negative electrode, and conducts a lithium ion.