3. In $\triangle A B C$, $O$ is the circumcenter of $\triangle A B C$, $D$ is the midpoint of $A C$, and it satisfies $\overrightarrow{O B} \cdot \overrightarrow{O D}=\overrightarrow{O A} \cdot \overrightarrow{O C}$. Then the minimum value of $\frac{\sin B}{\tan A}+\frac{\sin B}{\tan C}$ is $\qquad$ .
3. 1 .

Let the circumradius of $\triangle A B C$ be $R$.
Notice that, $O$ is the circumcenter of $\triangle A B C$.
$$
\begin{array}{l}
\text { By } \overrightarrow{O B} \cdot \overrightarrow{O D}=\overrightarrow{O A} \cdot \overrightarrow{O C} \\
\Rightarrow \overrightarrow{O B} \cdot(\overrightarrow{O A}+\overrightarrow{O C})=2 \overrightarrow{O A} \cdot \overrightarrow{O C} \\
\Rightarrow 2 R^{2} \cos 2 B=R^{2} \cos 2 A+R^{2} \cos 2 C \\
\Rightarrow 2 \cos 2 B=\cos 2 A+\cos 2 C \\
\Rightarrow \cos 2 B-\cos 2 A=\cos 2 C-\cos 2 B \\
\Rightarrow \sin C \cdot \sin (B-A)=\sin A \cdot \sin (C-B) \\
\Rightarrow b c \cos A+a b \cos C=2 a c \cos B \\
\Rightarrow a^{2}+c^{2}=2 b^{2} . \\
\text { And } \frac{\sin B}{\tan A}+\frac{\sin B}{\tan C}=\frac{b^{2}}{a c} \geqslant 1, \text { with equality if and only if } a=c
\end{array}
$$

Therefore, the minimum value sought is 1.