The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over
Let us consider an analytic function $f$ defined in the whole complex plane which has infinitely many zeros. Let us restrict the function to the interval $(0,1)$ as follow: $g(t)=f(1-2t)$. I look for the number of roots of $g$ in $(0,1)$.
My question is: What I can say for the case of $g$ defined by using $f$ in $(0,1)$.