Conjugate momenta and equations of motion

Suppose a dynamical system has the Lagrangian

$$L=(\dot q_1)^2+(\dot q_2)^2+(q_1)^2+\dot q_1\dot q_2.$$

If $p_1$ and $p_2$ are momenta conjugate to $q_1$ and $q_2$, respectively, then which of the following statements is true?