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# On the Error Term in Duke's Estimate for the Average Special Value of $L$-Functions

Let $\FF$ be an orthonormal basis for weight $2$ cusp forms of level $N$. We show that various weighted averages of special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4 \pi c + O(N^{-1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error term of $O(N^{-1/2}\log N)$.
 MSC Classifications: 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F11 - Holomorphic modular forms of integral weight