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Properties

Label 8-425e4-1.1-c1e4-0-4
Degree 88
Conductor 3262539062532625390625
Sign 11
Analytic cond. 132.636132.636
Root an. cond. 1.842181.84218
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 8·4-s + 16·6-s − 4·7-s − 12·8-s + 4·9-s − 4·11-s − 32·12-s + 16·14-s + 15·16-s − 12·17-s − 16·18-s − 8·19-s + 16·21-s + 16·22-s − 12·23-s + 48·24-s + 4·27-s − 32·28-s + 4·29-s − 12·31-s − 16·32-s + 16·33-s + 48·34-s + 32·36-s + 20·37-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s − 1.51·7-s − 4.24·8-s + 4/3·9-s − 1.20·11-s − 9.23·12-s + 4.27·14-s + 15/4·16-s − 2.91·17-s − 3.77·18-s − 1.83·19-s + 3.49·21-s + 3.41·22-s − 2.50·23-s + 9.79·24-s + 0.769·27-s − 6.04·28-s + 0.742·29-s − 2.15·31-s − 2.82·32-s + 2.78·33-s + 8.23·34-s + 16/3·36-s + 3.28·37-s + ⋯

Functional equation

Λ(s)=((58⋅174)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((58⋅174)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 58⋅1745^{8} \cdot 17^{4}
Sign: 11
Analytic conductor: 132.636132.636
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 58⋅174, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad5 1 1
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
good2D4×C2D_4\times C_2 1+p2T+p3T2+3p2T3+17T4+3p3T5+p5T6+p5T7+p4T8 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8}
3D4×C2D_4\times C_2 1+4T+4pT2+28T3+56T4+28pT5+4p3T6+4p3T7+p4T8 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
7D4×C2D_4\times C_2 1+4T+12T2+44T3+120T4+44pT5+12p2T6+4p3T7+p4T8 1 + 4 T + 12 T^{2} + 44 T^{3} + 120 T^{4} + 44 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+4T+12T2+60T3+184T4+60pT5+12p2T6+4p3T7+p4T8 1 + 4 T + 12 T^{2} + 60 T^{3} + 184 T^{4} + 60 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
13C22C_2^2 (1+24T2+p2T4)2 ( 1 + 24 T^{2} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1+8T+32T2+184T3+1042T4+184pT5+32p2T6+8p3T7+p4T8 1 + 8 T + 32 T^{2} + 184 T^{3} + 1042 T^{4} + 184 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+12T+68T2+316T3+1496T4+316pT5+68p2T6+12p3T7+p4T8 1 + 12 T + 68 T^{2} + 316 T^{3} + 1496 T^{4} + 316 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−4T+22T2−244T3+930T4−244pT5+22p2T6−4p3T7+p4T8 1 - 4 T + 22 T^{2} - 244 T^{3} + 930 T^{4} - 244 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1+12T+108T2+804T3+5112T4+804pT5+108p2T6+12p3T7+p4T8 1 + 12 T + 108 T^{2} + 804 T^{3} + 5112 T^{4} + 804 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
37D4×C2D_4\times C_2 1−20T+150T2−500T3+1250T4−500pT5+150p2T6−20p3T7+p4T8 1 - 20 T + 150 T^{2} - 500 T^{3} + 1250 T^{4} - 500 p T^{5} + 150 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
41C2C_2×\timesC22C_2^2 (1−6T+pT2)2(1+16T+128T2+16pT3+p2T4) ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )
43D4×C2D_4\times C_2 1+8T+32T2+376T3+4402T4+376pT5+32p2T6+8p3T7+p4T8 1 + 8 T + 32 T^{2} + 376 T^{3} + 4402 T^{4} + 376 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
47D4D_{4} (1+16T+150T2+16pT3+p2T4)2 ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−2T+2T2−2pT3+p2T4)2 ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
59C22C_2^2×\timesC22C_2^2 (1−20T+200T2−20pT3+p2T4)(1+20T+200T2+20pT3+p2T4) ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )
61D4×C2D_4\times C_2 1+50T2−720T3+1250T4−720pT5+50p2T6+p4T8 1 + 50 T^{2} - 720 T^{3} + 1250 T^{4} - 720 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−220T2+20566T4−220p2T6+p4T8 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−20T+100T2+20pT3−23400T4+20p2T5+100p2T6−20p3T7+p4T8 1 - 20 T + 100 T^{2} + 20 p T^{3} - 23400 T^{4} + 20 p^{2} T^{5} + 100 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
73D4×C2D_4\times C_2 1+98T2+672T3+4802T4+672pT5+98p2T6+p4T8 1 + 98 T^{2} + 672 T^{3} + 4802 T^{4} + 672 p T^{5} + 98 p^{2} T^{6} + p^{4} T^{8}
79D4×C2D_4\times C_2 1−4T+12T2+836T3−3400T4+836pT5+12p2T6−4p3T7+p4T8 1 - 4 T + 12 T^{2} + 836 T^{3} - 3400 T^{4} + 836 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+16T+128T2+1264T3+12466T4+1264pT5+128p2T6+16p3T7+p4T8 1 + 16 T + 128 T^{2} + 1264 T^{3} + 12466 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
89D4×C2D_4\times C_2 1−224T2+27874T4−224p2T6+p4T8 1 - 224 T^{2} + 27874 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−4T+54T2+1388T3−4894T4+1388pT5+54p2T6−4p3T7+p4T8 1 - 4 T + 54 T^{2} + 1388 T^{3} - 4894 T^{4} + 1388 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−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

−8.461712094088775646650720759822, −8.395170124393985841844533636386, −8.247876836264968698360555225332, −8.043099558011723850337980177806, −7.84276778428535037952558298962, −7.41365566151151161894465217570, −6.83732561576051257122083323241, −6.74139466625076497267501010283, −6.72645989616945193200399518182, −6.51525938650821944907948057307, −6.25003174254374457773934081906, −5.83150170105886121986071238743, −5.78602528569735038141163518109, −5.74969100367221134875452148429, −4.96999559640665771319244717240, −4.95124961730577736204645377171, −4.62157351961607601781014946113, −4.05241045658196203350523733790, −3.99425172762489601557091065438, −3.42004887698769473364810613663, −3.01217169934508413766871457344, −2.39289955090906246927740323385, −2.35485851093939593475873485339, −2.01830063658283705206385857218, −1.33682807989344830247393465810, 0, 0, 0, 0, 1.33682807989344830247393465810, 2.01830063658283705206385857218, 2.35485851093939593475873485339, 2.39289955090906246927740323385, 3.01217169934508413766871457344, 3.42004887698769473364810613663, 3.99425172762489601557091065438, 4.05241045658196203350523733790, 4.62157351961607601781014946113, 4.95124961730577736204645377171, 4.96999559640665771319244717240, 5.74969100367221134875452148429, 5.78602528569735038141163518109, 5.83150170105886121986071238743, 6.25003174254374457773934081906, 6.51525938650821944907948057307, 6.72645989616945193200399518182, 6.74139466625076497267501010283, 6.83732561576051257122083323241, 7.41365566151151161894465217570, 7.84276778428535037952558298962, 8.043099558011723850337980177806, 8.247876836264968698360555225332, 8.395170124393985841844533636386, 8.461712094088775646650720759822

Graph of the ZZ-function along the critical line