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Given a modular form $f$ , the corresponding $L$-function $L(s,f)$ satisfies a functional equation as $s\mapsto 1-s$. One can view bounds of the form $L(1/2, f ) \ll C( f )^{\alpha+\varepsilon}$ as improvements towards the $\alpha = 0$ case, the Lindelof Hypothesis. The functional equation together with the Phragmen Lindelof theorem give the bound for $\alpha =1/4$. This is called the convexity bound, and any result with $\alpha < 1/4$ is called the subconvexity result. The Lindelof Hypothesis is a consequence of the Riemann Hypothesis for $L(s,f)$ and thus are expected to be true for all $L$-functions with a functional equation and an Euler product. We prove a subconvexity bound for $L(1/2; f ,\chi)$ in the conductor aspect where $f$ is a half integral weight modular form. In this case the L function does not possess an Euler product.