Geodesic
A uniform asymptotic formula for the second moment of primitive $L$-functions on the critical line
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 20-53

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We prove an asymptotic formula for the second moment of primitive $L$-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist. 
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     author = {Olga G. Balkanova and Dmitry A. Frolenkov},
     title = {A uniform asymptotic formula for the second moment of primitive $L$-functions on the critical line},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {20--53},
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     volume = {294},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_294_a1/}
}
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Olga G. Balkanova; Dmitry A. Frolenkov. A uniform asymptotic formula for the second moment of primitive $L$-functions on the critical line. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 20-53. http://geodesic.mathdoc.fr/item/TM_2016_294_a1/