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The coefficients of $\zeta (s)L(s,\chi )$ are multiplicative. On $p^k$, their values are $k+1$ when $\chi (p)=1$, alternating one and zero when $\chi (p)=-1$, and one when $\chi (p)=0$. In particular $a_{m^2}\ge 1$. Their logarithmic Euler coefficients $\Lambda (n)(1+\chi (n))$ are also nonnegative, supporting uniqueness of a possible exceptional real Dirichlet zero.

Put $D_0=|D|$. The reference to part (c) in the printed hint is a reference to the positive-coefficient function from part (b). For $\sigma >1$ close to one, the preceding positivity and the supplied partial-fraction expansion giveAll omitted zero terms have nonnegative real parts because their real parts are at most one. The $O(1)$ zeta-pole remainder is included in $C\log D_0$, increasing the absolute constant if needed; a nonprincipal primitive real conductor of a Dirichlet character is at least three.

Suppose there were two real zeros, counted with multiplicity, with $\beta \ge 1-c/\log D_0$. Set $\sigma =1+a/\log D_0$. Division by $\log D_0$ givesChoose $C\ge 1$, $a=1/(4C)$ and $c=a/4$. The right side is strictly negative. ThusThis is the uniqueness of a possible exceptional real Dirichlet zero. It proves uniqueness, rather than existence of such a zero.