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## The mean value of symmetric square L-functions

Algebra and Number Theory. 2018. Vol. 12. P. 35-59.
Balkanova O., Frolenkov D.

We study the first moment of symmetric-square $L$-functions at the critical point  in the weight aspect. Asymptotics with the best known error term $O(k^{-1/2})$ were obtained independently by Fomenko in 2005 and by Sun in 2013.
We prove that there is an extra main term of size $k^{-1/2}$ in the asymptotic formula and show that the remainder term decays exponentially in $k$. The twisted first moment was evaluated asymptotically by Ng with the error bounded by $lk^{-1/2+\epsilon}$. We improve the error bound to $l^{5/6+\epsilon}k^{-1/2+\epsilon}$ unconditionally and to $l^{1/2+\epsilon}k^{-1/2}$ under the Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions.