= Solution
The <global partial-fraction expansion of the zeta logarithmic derivative>, with zeros counted with multiplicity and paired summands retained, is
$$
\frac{\zeta'}{\zeta}(s)=B-\frac1s-\frac1{s-1}
+\frac12\log\pi-\frac12\frac{\Gamma'(s/2)}{\Gamma(s/2)}
+\sum_\rho\left(\frac1{s-\rho}+\frac1\rho\right),
\qquad B=\frac{\xi'(0)}{\xi(0)}.
$$
For a nonprincipal <primitive Dirichlet character> $\chi$ of <Dirichlet conductor> $q$, put $a=0$ or $1$ according as $\chi(-1)=1$ or $-1$. Its <completed Dirichlet L-function> is
$$
\mathcal L_\chi(s)=\left(\frac q\pi\right)^{(s+a)/2}
\Gamma\left(\frac{s+a}{2}\right)L(s,\chi).
$$
The <global partial-fraction expansion of a Dirichlet L-function logarithmic derivative> is
$$
\boxed{\frac{L'}{L}(s,\chi)=B_\chi-\frac12\log\frac q\pi
-\frac12\frac{\Gamma'((s+a)/2)}{\Gamma((s+a)/2)}
+\sum_{\rho_\chi}\left(\frac1{s-\rho_\chi}+\frac1{\rho_\chi}\right),}
$$
where $B_\chi=\mathcal L_\chi'(0)/\mathcal L_\chi(0)$ and the zeros are the nontrivial zeros of the completion. These expansions follow from genus-one <Hadamard factorization>; the combined terms converge locally normally. The primitive principal <Dirichlet character> has <Dirichlet conductor> one and gives the zeta formula. If a <Dirichlet character> modulo $Q$ is induced by a <primitive Dirichlet character> $\chi^*$ of <Dirichlet conductor> $q$, its missing Euler factors add
$$
\sum_{\substack{p\mid Q\\p\nmid q}}
\frac{\chi^*(p)\log p}{p^s-\chi^*(p)}
$$
to the primitive <logarithmic derivative>. This correction is $O(\log Q)$ on the real interval $1<\sigma\leq2$.
The form needed here is <Landau theorem for two real Dirichlet characters>: there is an absolute $b>0$ such that, if $\chi_1,\chi_2$ are real primitive nonprincipal <Dirichlet characters> of <Dirichlet conductors> $q_1,q_2$ and their product is nonprincipal, then
$$
\boxed{L(s,\chi_1)L(s,\chi_2)\text{ has at most one real zero in }
1-\frac b{\log(q_1q_2)}<s<1,}
$$
counting multiplicity. In particular, two distinct primitive <real Dirichlet characters> cannot each have a zero in that interval.
For the proof, induce the <Dirichlet characters> to a common modulus and consider
$$
D(s)=\zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2).
$$
The last factor can be imprimitive but is nonprincipal. For $\sigma>1$, logarithmic differentiation of the <Euler products> gives
$$
-\frac{D'}D(\sigma)=\sum_{n\geq1}
\frac{\Lambda(n)(1+\chi_1(n))(1+\chi_2(n))}{n^\sigma}\geq0.
$$
This positivity is the reason to include all four factors: each <real Dirichlet character> takes values in $\{0,1,-1\}$, including at ramified <primes>.
The real partial-fraction identity for a primitive completion has no residual real constant. Indeed, after taking real parts in its Hadamard derivative, the constant is determined to be zero by the functional-equation symmetry of its modulus across the critical line; the zero kernels there cancel in reflected pairs. Consequently
$$
\Re\frac{L'}L(s,\chi)
=\sum_{\rho_\chi}\Re\frac1{s-\rho_\chi}
-\frac12\log\frac q\pi
-\frac12\Re\frac{\Gamma'((s+a)/2)}{\Gamma((s+a)/2)}.
$$
For $1<\sigma\leq2$ all the zero kernels are nonnegative, and the gamma term is bounded. The missing Euler factors have the previously bounded correction. Combining the four <logarithmic derivatives>, and retaining any two candidate real zeros $\beta_1,\beta_2$ of $L(\chi_1)L(\chi_2)$, therefore gives
$$
0\leq-\frac{D'}D(\sigma)
\leq\frac1{\sigma-1}
-\frac1{\sigma-\beta_1}-\frac1{\sigma-\beta_2}
+C\log(q_1q_2).
$$
Zeros in any other factors only decrease this upper bound. The sole <pole> term comes from zeta.
Put $H=\log(q_1q_2)$ and choose $a>0$ so small that $Ca<1/10$. If both $1-\beta_j<a/(4H)$, take $\sigma=1+a/H$. The upper bound is then at most
$$
H\left(\frac1a-\frac2{a+a/4}+C\right)
=H\left(C-\frac3{5a}\right)<0,
$$
a contradiction. Taking $b=a/4$ proves the theorem, including the multiplicity assertion.
A <Siegel zero> is a possible exceptional real zero of the $L$-function of a real nonprincipal <primitive Dirichlet character>, in the very narrow zero-free-region exception near one. Fix a sufficiently small absolute threshold $\eta>0$ and call a zero exceptional here when
$$
1-\frac\eta{\log q}<\beta<1.
$$
Take $\eta<b/4$ and also smaller than the constant in the <classical zero-free region for Dirichlet L-functions>. Such a zero, if present, is real and simple; its existence is not asserted. This choice fixes the constants consistently for the requested separation conclusions.
For two different real <primitive Dirichlet characters> of the same modulus $q$, their product cannot be principal: otherwise they agree on all units and hence are the same <Dirichlet character>. Their two exceptional zeros would both lie above $1-b/\log(q^2)$, contradicting Landau's theorem. Thus \b[at most one real <primitive Dirichlet character> modulo $q$ has an exceptional zero].
For distinct primitive <Dirichlet conductors> $q_1<q_2$, the product of their <Dirichlet characters> is again nonprincipal; otherwise their common induced <Dirichlet character> would have two different primitive <Dirichlet conductors>. Suppose $q_2\leq q_1^2$. Then
$$
\log(q_1q_2)\leq3\log q_1,\qquad
1-\beta_j<\frac\eta{\log q_j}\leq\frac\eta{\log q_1}
<\frac b{\log(q_1q_2)}\quad(j=1,2).
$$
Again both zeros violate the two-zero prohibition. Therefore the <sparse conductors of exceptional real Dirichlet zeros> obey
$$
\boxed{q_{j+1}>q_j^2.}
$$
The quadratic spacing uses a fixed sufficiently small exceptional-zero constant; it does not claim this conclusion for an arbitrarily enlarged neighbourhood of one.
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