For a sinusoidal input $x(t)=\sin(\omega_0t+\varphi)$, the output of a real-valued LTI system is given by
where $H(j\omega)$ is the system's frequency response. The amplitude of the output is scaled by the magnitude of $H(j\omega)$ evaluated at $\omega=\omega_0$, and the phase is shifted by the argument of $H(j\omega_0)$.
If the input is a constant ($x(t)=c$), then in analogy with $(1)$, the output is given by
Since the system is linear, we can use superposition, and compute the output for each input component and then sum up all the individual components. So the output corresponding to the input $x(t)=c+\sin(\omega_0t+\varphi)$ is given by
So there is no contradiction between a gain of $6$ at frequency $\omega_0$ and a gain of $0$ at frequency $0$ (DC). From the given input and output you can deduce the values of $H(0)$ and $H(\omega_0)$.