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Here is the example I always give to the students. Take a symmetrical random variable $X$ with zero mean (such as normal). Then $EX=0$ and $EX^3=0$. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$