Properties

Label 8-294e4-1.1-c5e4-0-3
Degree 88
Conductor 74711820967471182096
Sign 11
Analytic cond. 4.94346×1064.94346\times 10^{6}
Root an. cond. 6.866796.86679
Motivic weight 55
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  + 8·2-s − 18·3-s + 16·4-s − 108·5-s − 144·6-s − 128·8-s + 81·9-s − 864·10-s − 124·11-s − 288·12-s + 1.44e3·13-s + 1.94e3·15-s − 1.02e3·16-s + 1.26e3·17-s + 648·18-s − 360·19-s − 1.72e3·20-s − 992·22-s − 6.52e3·23-s + 2.30e3·24-s + 6.71e3·25-s + 1.15e4·26-s + 1.45e3·27-s + 1.41e4·29-s + 1.55e4·30-s − 5.90e3·31-s − 2.04e3·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.93·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s − 2.73·10-s − 0.308·11-s − 0.577·12-s + 2.36·13-s + 2.23·15-s − 16-s + 1.05·17-s + 0.471·18-s − 0.228·19-s − 0.965·20-s − 0.436·22-s − 2.57·23-s + 0.816·24-s + 2.14·25-s + 3.34·26-s + 0.384·27-s + 3.13·29-s + 3.15·30-s − 1.10·31-s − 0.353·32-s + ⋯

Functional equation

Λ(s)=((243478)s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((243478)s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 2434782^{4} \cdot 3^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 4.94346×1064.94346\times 10^{6}
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 243478, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 1.4736369261.473636926
L(12)L(\frac12) \approx 1.4736369261.473636926
L(72)L(\frac{7}{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 (1p2T+p4T2)2 ( 1 - p^{2} T + p^{4} T^{2} )^{2}
3C2C_2 (1+p2T+p4T2)2 ( 1 + p^{2} T + p^{4} T^{2} )^{2}
7 1 1
good5D4×C2D_4\times C_2 1+108T+4948T2+50328T31110969T4+50328p5T5+4948p10T6+108p15T7+p20T8 1 + 108 T + 4948 T^{2} + 50328 T^{3} - 1110969 T^{4} + 50328 p^{5} T^{5} + 4948 p^{10} T^{6} + 108 p^{15} T^{7} + p^{20} T^{8}
11D4×C2D_4\times C_2 1+124T278818T23460592T3+58136364859T43460592p5T5278818p10T6+124p15T7+p20T8 1 + 124 T - 278818 T^{2} - 3460592 T^{3} + 58136364859 T^{4} - 3460592 p^{5} T^{5} - 278818 p^{10} T^{6} + 124 p^{15} T^{7} + p^{20} T^{8}
13D4D_{4} (1720T+673736T2720p5T3+p10T4)2 ( 1 - 720 T + 673736 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} )^{2}
17D4×C2D_4\times C_2 11260T1209092T2+54207720T3+3551327269215T4+54207720p5T51209092p10T61260p15T7+p20T8 1 - 1260 T - 1209092 T^{2} + 54207720 T^{3} + 3551327269215 T^{4} + 54207720 p^{5} T^{5} - 1209092 p^{10} T^{6} - 1260 p^{15} T^{7} + p^{20} T^{8}
19D4×C2D_4\times C_2 1+360T4647630T262988480T3+16369957784699T462988480p5T54647630p10T6+360p15T7+p20T8 1 + 360 T - 4647630 T^{2} - 62988480 T^{3} + 16369957784699 T^{4} - 62988480 p^{5} T^{5} - 4647630 p^{10} T^{6} + 360 p^{15} T^{7} + p^{20} T^{8}
23D4×C2D_4\times C_2 1+6524T+19335014T2+2937183088pT3+423716043763p2T4+2937183088p6T5+19335014p10T6+6524p15T7+p20T8 1 + 6524 T + 19335014 T^{2} + 2937183088 p T^{3} + 423716043763 p^{2} T^{4} + 2937183088 p^{6} T^{5} + 19335014 p^{10} T^{6} + 6524 p^{15} T^{7} + p^{20} T^{8}
29D4D_{4} (17088T+48216146T27088p5T3+p10T4)2 ( 1 - 7088 T + 48216146 T^{2} - 7088 p^{5} T^{3} + p^{10} T^{4} )^{2}
31D4×C2D_4\times C_2 1+5904T23971190T2+9269894016T3+1643220565216419T4+9269894016p5T523971190p10T6+5904p15T7+p20T8 1 + 5904 T - 23971190 T^{2} + 9269894016 T^{3} + 1643220565216419 T^{4} + 9269894016 p^{5} T^{5} - 23971190 p^{10} T^{6} + 5904 p^{15} T^{7} + p^{20} T^{8}
37D4×C2D_4\times C_2 16040T74621402T2+166612868480T3+5005514179302955T4+166612868480p5T574621402p10T66040p15T7+p20T8 1 - 6040 T - 74621402 T^{2} + 166612868480 T^{3} + 5005514179302955 T^{4} + 166612868480 p^{5} T^{5} - 74621402 p^{10} T^{6} - 6040 p^{15} T^{7} + p^{20} T^{8}
41D4D_{4} (1+17388T+216792980T2+17388p5T3+p10T4)2 ( 1 + 17388 T + 216792980 T^{2} + 17388 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4D_{4} (1+608T+164053110T2+608p5T3+p10T4)2 ( 1 + 608 T + 164053110 T^{2} + 608 p^{5} T^{3} + p^{10} T^{4} )^{2}
47D4×C2D_4\times C_2 1648pT+298147738T2110633159232pT3+130837578639633603T4110633159232p6T5+298147738p10T6648p16T7+p20T8 1 - 648 p T + 298147738 T^{2} - 110633159232 p T^{3} + 130837578639633603 T^{4} - 110633159232 p^{6} T^{5} + 298147738 p^{10} T^{6} - 648 p^{16} T^{7} + p^{20} T^{8}
53D4×C2D_4\times C_2 1+3964T578718526T2959126126096T3+171890491073351707T4959126126096p5T5578718526p10T6+3964p15T7+p20T8 1 + 3964 T - 578718526 T^{2} - 959126126096 T^{3} + 171890491073351707 T^{4} - 959126126096 p^{5} T^{5} - 578718526 p^{10} T^{6} + 3964 p^{15} T^{7} + p^{20} T^{8}
59D4×C2D_4\times C_2 1+40752T180305678T2+16756512663168T3+1690986043017054027T4+16756512663168p5T5180305678p10T6+40752p15T7+p20T8 1 + 40752 T - 180305678 T^{2} + 16756512663168 T^{3} + 1690986043017054027 T^{4} + 16756512663168 p^{5} T^{5} - 180305678 p^{10} T^{6} + 40752 p^{15} T^{7} + p^{20} T^{8}
61D4×C2D_4\times C_2 11368T1543594872T2+196617586608T3+1673542562621074391T4+196617586608p5T51543594872p10T61368p15T7+p20T8 1 - 1368 T - 1543594872 T^{2} + 196617586608 T^{3} + 1673542562621074391 T^{4} + 196617586608 p^{5} T^{5} - 1543594872 p^{10} T^{6} - 1368 p^{15} T^{7} + p^{20} T^{8}
67D4×C2D_4\times C_2 116224T549961574T2+30615831207936T31516953966778043781T4+30615831207936p5T5549961574p10T616224p15T7+p20T8 1 - 16224 T - 549961574 T^{2} + 30615831207936 T^{3} - 1516953966778043781 T^{4} + 30615831207936 p^{5} T^{5} - 549961574 p^{10} T^{6} - 16224 p^{15} T^{7} + p^{20} T^{8}
71D4D_{4} (1+3204T212201462T2+3204p5T3+p10T4)2 ( 1 + 3204 T - 212201462 T^{2} + 3204 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1+23976T+506144208T297760673100368T35484607792703379793T497760673100368p5T5+506144208p10T6+23976p15T7+p20T8 1 + 23976 T + 506144208 T^{2} - 97760673100368 T^{3} - 5484607792703379793 T^{4} - 97760673100368 p^{5} T^{5} + 506144208 p^{10} T^{6} + 23976 p^{15} T^{7} + p^{20} T^{8}
79D4×C2D_4\times C_2 11040pT1035403070T21696818106880pT3+30377412596963147299T41696818106880p6T51035403070p10T61040p16T7+p20T8 1 - 1040 p T - 1035403070 T^{2} - 1696818106880 p T^{3} + 30377412596963147299 T^{4} - 1696818106880 p^{6} T^{5} - 1035403070 p^{10} T^{6} - 1040 p^{16} T^{7} + p^{20} T^{8}
83D4D_{4} (1173736T+14732805782T2173736p5T3+p10T4)2 ( 1 - 173736 T + 14732805782 T^{2} - 173736 p^{5} T^{3} + p^{10} T^{4} )^{2}
89D4×C2D_4\times C_2 1+200556T+19064326876T2+2003607258829272T3+ 1 + 200556 T + 19064326876 T^{2} + 2003607258829272 T^{3} + 19 ⁣ ⁣3519\!\cdots\!35T4+2003607258829272p5T5+19064326876p10T6+200556p15T7+p20T8 T^{4} + 2003607258829272 p^{5} T^{5} + 19064326876 p^{10} T^{6} + 200556 p^{15} T^{7} + p^{20} T^{8}
97D4D_{4} (1251928T+32348722272T2251928p5T3+p10T4)2 ( 1 - 251928 T + 32348722272 T^{2} - 251928 p^{5} T^{3} + p^{10} T^{4} )^{2}
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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

−7.63111350276851282839529306859, −7.58033851108019026727045520910, −6.87432692669783178625408153198, −6.85144687383743232668091520067, −6.46747154302176764150305047309, −6.23335823569125001920137255739, −6.02922364482775022172083533449, −5.81885558169677874373478035254, −5.69018738939597738078411138806, −5.05909528642798878902088862602, −4.88470861381797690758866630384, −4.75816470690841193528849070244, −4.50958013114299861744887360528, −4.04853548720884862749081578565, −3.81783796757596073178725100313, −3.48574176551513432328104320130, −3.44172828692050778258738652739, −3.28653292807623554903241064502, −2.65557472819995092016172701871, −2.16989529920840359056549155876, −1.83577275211195568490860382596, −1.09198641351113775959050559666, −0.978045514965047791858217735623, −0.55655967611740390127177039663, −0.19472807488905400749879755380, 0.19472807488905400749879755380, 0.55655967611740390127177039663, 0.978045514965047791858217735623, 1.09198641351113775959050559666, 1.83577275211195568490860382596, 2.16989529920840359056549155876, 2.65557472819995092016172701871, 3.28653292807623554903241064502, 3.44172828692050778258738652739, 3.48574176551513432328104320130, 3.81783796757596073178725100313, 4.04853548720884862749081578565, 4.50958013114299861744887360528, 4.75816470690841193528849070244, 4.88470861381797690758866630384, 5.05909528642798878902088862602, 5.69018738939597738078411138806, 5.81885558169677874373478035254, 6.02922364482775022172083533449, 6.23335823569125001920137255739, 6.46747154302176764150305047309, 6.85144687383743232668091520067, 6.87432692669783178625408153198, 7.58033851108019026727045520910, 7.63111350276851282839529306859

Graph of the ZZ-function along the critical line