  Starting from the Davenport-Heilbronn function equation: $f(s) = X(s)
f(1-s)$, we discover the four properties of the meromorphic function $X(s)$
defined as the ratio of the Davenport-Heilbronn functions: $\frac{f(s)}{f(1-s)}
= X(s)$, and three corresponding lemmas. For the first time, we propose to
study the distribution of the non-trivial zeros of the Davenport-Heilbronn
function by exploring the monotonicity of the similarity ratio $\left|
\frac{f(s)}{f(1-s)} \right|$. We point out that for the $f(s)$ which satisfies
the Davenport-Heilbronn function equation, the existence of non-trivial zeros
outside of the critical line $\{ s_n | \sigma \neq 1/2 \}$ presents two
puzzles: 1) $f(s_n) \neq f(1 - s_n)$; 2) the existence of non-trivial zeros $\{
s_n | \sigma \neq 1/2 \}$ is in contradiction of the monotonicity of the
similar ratio $\left| \frac{f(s)}{f(1-s)}\right|$.
