Properties

Label 8-50e4-1.1-c23e4-0-0
Degree 88
Conductor 62500006250000
Sign 11
Analytic cond. 7.89072×1087.89072\times 10^{8}
Root an. cond. 12.946112.9461
Motivic weight 2323
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.38e6·4-s + 2.00e11·9-s − 1.38e12·11-s + 5.27e13·16-s + 1.94e15·19-s + 8.07e16·29-s − 1.09e17·31-s − 1.68e18·36-s − 8.54e18·41-s + 1.16e19·44-s + 6.48e19·49-s − 8.03e19·59-s + 1.60e21·61-s − 2.95e20·64-s + 1.43e21·71-s − 1.62e22·76-s + 1.41e22·79-s + 1.31e22·81-s + 3.96e20·89-s − 2.77e23·99-s + 9.33e22·101-s + 1.20e24·109-s − 6.77e23·116-s − 2.23e24·121-s + 9.19e23·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 2.12·9-s − 1.46·11-s + 3/4·16-s + 3.82·19-s + 1.22·29-s − 0.774·31-s − 2.12·36-s − 2.42·41-s + 1.46·44-s + 2.37·49-s − 0.347·59-s + 4.72·61-s − 1/2·64-s + 0.737·71-s − 3.82·76-s + 2.12·79-s + 1.48·81-s + 0.0151·89-s − 3.11·99-s + 0.832·101-s + 4.48·109-s − 1.22·116-s − 2.49·121-s + 0.774·124-s + ⋯

Functional equation

Λ(s)=(6250000s/2ΓC(s)4L(s)=(Λ(24s)\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}
Λ(s)=(6250000s/2ΓC(s+23/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 62500006250000    =    24582^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 7.89072×1087.89072\times 10^{8}
Root analytic conductor: 12.946112.9461
Motivic weight: 2323
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 6250000, ( :23/2,23/2,23/2,23/2), 1)(8,\ 6250000,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)

Particular Values

L(12)L(12) \approx 0.77333136340.7733313634
L(12)L(\frac12) \approx 0.77333136340.7733313634
L(252)L(\frac{25}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 (1+p22T2)2 ( 1 + p^{22} T^{2} )^{2}
5 1 1
good3D4×C2D_4\times C_2 122271434820p2T2+458001548506188742p10T422271434820p48T6+p92T8 1 - 22271434820 p^{2} T^{2} + 458001548506188742 p^{10} T^{4} - 22271434820 p^{48} T^{6} + p^{92} T^{8}
7D4×C2D_4\times C_2 11323849609095360660p2T2+ 1 - 1323849609095360660 p^{2} T^{2} + 20 ⁣ ⁣0220\!\cdots\!02p6T41323849609095360660p48T6+p92T8 p^{6} T^{4} - 1323849609095360660 p^{48} T^{6} + p^{92} T^{8}
11D4D_{4} (1+62935415616pT+ ( 1 + 62935415616 p T + 15 ⁣ ⁣8615\!\cdots\!86p2T2+62935415616p24T3+p46T4)2 p^{2} T^{2} + 62935415616 p^{24} T^{3} + p^{46} T^{4} )^{2}
13D4×C2D_4\times C_2 1 1 - 25 ⁣ ⁣6025\!\cdots\!60T2+ T^{2} + 20 ⁣ ⁣2220\!\cdots\!22p2T4 p^{2} T^{4} - 25 ⁣ ⁣6025\!\cdots\!60p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
17D4×C2D_4\times C_2 1 1 - 63 ⁣ ⁣4063\!\cdots\!40T2+ T^{2} + 61 ⁣ ⁣4261\!\cdots\!42p2T4 p^{2} T^{4} - 63 ⁣ ⁣4063\!\cdots\!40p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
19D4D_{4} (1970438534616600T+ ( 1 - 970438534616600 T + 38 ⁣ ⁣2238\!\cdots\!22pT2970438534616600p23T3+p46T4)2 p T^{2} - 970438534616600 p^{23} T^{3} + p^{46} T^{4} )^{2}
23D4×C2D_4\times C_2 1 1 - 62 ⁣ ⁣2062\!\cdots\!20T2+ T^{2} + 18 ⁣ ⁣7818\!\cdots\!78T4 T^{4} - 62 ⁣ ⁣2062\!\cdots\!20p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
29D4D_{4} (140394756391690180T+ ( 1 - 40394756391690180 T + 26 ⁣ ⁣7826\!\cdots\!78T240394756391690180p23T3+p46T4)2 T^{2} - 40394756391690180 p^{23} T^{3} + p^{46} T^{4} )^{2}
31D4D_{4} (1+54819002015519096T+ ( 1 + 54819002015519096 T + 33 ⁣ ⁣8633\!\cdots\!86T2+54819002015519096p23T3+p46T4)2 T^{2} + 54819002015519096 p^{23} T^{3} + p^{46} T^{4} )^{2}
37D4×C2D_4\times C_2 1 1 - 18 ⁣ ⁣0018\!\cdots\!00T2+ T^{2} + 23 ⁣ ⁣1823\!\cdots\!18T4 T^{4} - 18 ⁣ ⁣0018\!\cdots\!00p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
41D4D_{4} (1+4273739814696341076T+ ( 1 + 4273739814696341076 T + 22 ⁣ ⁣8622\!\cdots\!86T2+4273739814696341076p23T3+p46T4)2 T^{2} + 4273739814696341076 p^{23} T^{3} + p^{46} T^{4} )^{2}
43D4×C2D_4\times C_2 1 1 - 80 ⁣ ⁣8080\!\cdots\!80T2+ T^{2} + 38 ⁣ ⁣9838\!\cdots\!98T4 T^{4} - 80 ⁣ ⁣8080\!\cdots\!80p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
47D4×C2D_4\times C_2 1 1 - 34 ⁣ ⁣6034\!\cdots\!60T2+ T^{2} + 32 ⁣ ⁣5832\!\cdots\!58T4 T^{4} - 34 ⁣ ⁣6034\!\cdots\!60p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
53D4×C2D_4\times C_2 1 1 - 10 ⁣ ⁣4010\!\cdots\!40T2+ T^{2} + 64 ⁣ ⁣5864\!\cdots\!58T4 T^{4} - 10 ⁣ ⁣4010\!\cdots\!40p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
59D4D_{4} (1+40199521927491576840T+ ( 1 + 40199521927491576840 T + 66 ⁣ ⁣5866\!\cdots\!58T2+40199521927491576840p23T3+p46T4)2 T^{2} + 40199521927491576840 p^{23} T^{3} + p^{46} T^{4} )^{2}
61D4D_{4} (1 ( 1 - 80 ⁣ ⁣0480\!\cdots\!04T+ T + 37 ⁣ ⁣6637\!\cdots\!66T2 T^{2} - 80 ⁣ ⁣0480\!\cdots\!04p23T3+p46T4)2 p^{23} T^{3} + p^{46} T^{4} )^{2}
67D4×C2D_4\times C_2 1 1 - 14 ⁣ ⁣0014\!\cdots\!00T2+ T^{2} + 17 ⁣ ⁣3817\!\cdots\!38T4 T^{4} - 14 ⁣ ⁣0014\!\cdots\!00p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
71D4D_{4} (1 ( 1 - 71 ⁣ ⁣6471\!\cdots\!64T T - 15 ⁣ ⁣5415\!\cdots\!54T2 T^{2} - 71 ⁣ ⁣6471\!\cdots\!64p23T3+p46T4)2 p^{23} T^{3} + p^{46} T^{4} )^{2}
73D4×C2D_4\times C_2 1 1 - 82 ⁣ ⁣8082\!\cdots\!80T2+ T^{2} + 43 ⁣ ⁣7843\!\cdots\!78T4 T^{4} - 82 ⁣ ⁣8082\!\cdots\!80p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
79D4D_{4} (1 ( 1 - 70 ⁣ ⁣2070\!\cdots\!20T+ T + 10 ⁣ ⁣7810\!\cdots\!78T2 T^{2} - 70 ⁣ ⁣2070\!\cdots\!20p23T3+p46T4)2 p^{23} T^{3} + p^{46} T^{4} )^{2}
83D4×C2D_4\times C_2 1 1 - 17 ⁣ ⁣8017\!\cdots\!80T2+ T^{2} + 13 ⁣ ⁣3813\!\cdots\!38T4 T^{4} - 17 ⁣ ⁣8017\!\cdots\!80p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
89D4D_{4} (1 ( 1 - 19 ⁣ ⁣8019\!\cdots\!80T+ T + 11 ⁣ ⁣3811\!\cdots\!38T2 T^{2} - 19 ⁣ ⁣8019\!\cdots\!80p23T3+p46T4)2 p^{23} T^{3} + p^{46} T^{4} )^{2}
97D4×C2D_4\times C_2 1 1 - 72 ⁣ ⁣2072\!\cdots\!20T2+ T^{2} + 22 ⁣ ⁣5822\!\cdots\!58T4 T^{4} - 72 ⁣ ⁣2072\!\cdots\!20p46T6+p92T8 p^{46} T^{6} + p^{92} T^{8}
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   L(s)=p j=18(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.26267989440015201953483595839, −7.13604840584669986906481464201, −7.12509578474095856742407051083, −6.38558577042684424347480818659, −6.28342854487835865733797376851, −5.61245033721831569276492278109, −5.50023356415719037548480259330, −5.16056459698266726918008805581, −4.91063633745120909202895613576, −4.85980729291180296776594604691, −4.68273509971737381633048310653, −3.80144027922060055569715841749, −3.72074234548431133137519782365, −3.64688561076860561966045627701, −3.55381115303999432463981783678, −2.75033803246208929373718409054, −2.68844007942149560517496901297, −2.24880426914318431734225795638, −2.11637932776584669049868861872, −1.39185870470309711112928146636, −1.29979859032035994580253769868, −1.05606265038972972883681836332, −0.842146679937357100442094222784, −0.61085242407239084228608677716, −0.07813250672482166511139869778, 0.07813250672482166511139869778, 0.61085242407239084228608677716, 0.842146679937357100442094222784, 1.05606265038972972883681836332, 1.29979859032035994580253769868, 1.39185870470309711112928146636, 2.11637932776584669049868861872, 2.24880426914318431734225795638, 2.68844007942149560517496901297, 2.75033803246208929373718409054, 3.55381115303999432463981783678, 3.64688561076860561966045627701, 3.72074234548431133137519782365, 3.80144027922060055569715841749, 4.68273509971737381633048310653, 4.85980729291180296776594604691, 4.91063633745120909202895613576, 5.16056459698266726918008805581, 5.50023356415719037548480259330, 5.61245033721831569276492278109, 6.28342854487835865733797376851, 6.38558577042684424347480818659, 7.12509578474095856742407051083, 7.13604840584669986906481464201, 7.26267989440015201953483595839

Graph of the ZZ-function along the critical line