This answer to a related question on Math StackExchange indicates the following:
- "This leads you to the conclusion that products like $\zeta(s)^2$ or $L(s, \chi_1) L(s, \chi_2)$ should correspond to modular forms; and this is indeed the case, at least half of the time (there is a parity condition you have to impose)."
Question: How is the parity condition determined for modular forms corresponding to $L(s, \chi_1) L(s, \chi_2)$? Is it related to the parities of the two Dirichlet characters $\chi_1$ and $\chi_2$ (i.e. the values of $\chi_1(-1)$ and $\chi_2(-1)$)?
The condition for $L(\chi_1, s) L(\chi_2, s)$ to correspond to a modular form is that $\chi_1(-1) \chi_2(-1) = -1$. This is necessary, because otherwise the fudge factors in the functional equation don't match up (the shape of the functional equation of $L(\chi, s)$ depends on $\chi(-1)$). It is also sufficient because we can write down the corresponding modular forms directly (they are Eisenstein series).
More generally, for $L(s, \chi_1) L(s - n, \chi_2)$ to correspond to a modular form, it is necessary that $n \ge 0$ and $\chi_1(-1) \chi_2(-1) = (-1)^{n+1}$, and this is sufficient except in the bad case when $n = 1$ and $\chi_1 = \chi_2 = $ trivial character (in which case you don't get a modular form but something slightly weaker called a "quasimodular form").