What is the transfer function of a multiple feedback bandpass filter?
I finally found the right transfer function (thanks to slateraptor at eevblog):
$$\frac{V_o}{V_i}=-(\frac{R_3}{R_1+R_3})(\frac{j\omega R_2C}{-\omega^2(\frac{R_1R_3}{R_1+R_3})C^2R_2+j\omega(\frac{R_1R_3}{R_1+R_3})2C+1}) $$

I actually just forgot a \$\frac{1}{R_1}\$ factor in my first function.

This produces a much more logical bode diagram where only the 1kHz frequency is amplified: