$\zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2)$ is a Dirichlet series with nonnegative coefficients and $a_1 = 1$ (Davenport §21)

The Dirichlet series of $F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2)$ (Davenport §21). Let $\chi_1$ and $\chi_2$ be real (quadratic) Dirichlet characters modulo $q_1$ and $q_2$ respectively, and let $\chi_1\chi_2$ denote their product, a character modulo $q_1q_2$ (both characters lifted to the modulus $q_1q_2$). Then there are real numbers $a_n \ge 0$ with $a_1 = 1$ such that

the series converging absolutely for $\operatorname{Re} s > 1$.

The coefficients are $a = 1 * \chi_1 * \chi_2 * \chi_1\chi_2$ (Dirichlet convolution); $a$ is multiplicative, and at a prime power $p^k$ its value is the coefficient of $x^k$ in $\bigl((1-x)(1-\chi_1(p)x)(1-\chi_2(p)x)(1-\chi_1(p)\chi_2(p)x)\bigr)^{-1}$, which is nonnegative because $\chi_1(p),\chi_2(p) \in \{0,\pm1\}$ (equivalently, $\log F(s) = \sum_{p,m}(1+\chi_1(p^m))(1+\chi_2(p^m))\,p^{-ms}/m$ has nonnegative coefficients). This is the function $F(s)$ of Davenport §21 and the function $\zeta(s)f(s)$ of the second case of the proof of Siegel's theorem in Montgomery–Vaughan; the nonnegativity of its coefficients is what Estermann's lemma requires.

Formalization Note. Neither character is required to be primitive or non-principal; $\chi_1\chi_2$ is formed as the product of the two characters after changing both levels to $q_1q_2$ (Mathlib's changeLevel), so its value at $n$ is $\chi_1(n)\chi_2(n)$ for every integer $n$.