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Properties

Label 8-20e8-1.1-c1e4-0-5
Degree 88
Conductor 2560000000025600000000
Sign 11
Analytic cond. 104.075104.075
Root an. cond. 1.787181.78718
Motivic weight 11
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  − 4·2-s − 2·3-s + 8·4-s + 8·6-s − 8·8-s + 2·9-s + 6·11-s − 16·12-s + 4·13-s − 4·16-s + 8·17-s − 8·18-s + 6·19-s − 24·22-s + 16·24-s − 16·26-s + 4·27-s + 8·29-s + 4·31-s + 32·32-s − 12·33-s − 32·34-s + 16·36-s − 16·37-s − 24·38-s − 8·39-s + 4·43-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 4·4-s + 3.26·6-s − 2.82·8-s + 2/3·9-s + 1.80·11-s − 4.61·12-s + 1.10·13-s − 16-s + 1.94·17-s − 1.88·18-s + 1.37·19-s − 5.11·22-s + 3.26·24-s − 3.13·26-s + 0.769·27-s + 1.48·29-s + 0.718·31-s + 5.65·32-s − 2.08·33-s − 5.48·34-s + 8/3·36-s − 2.63·37-s − 3.89·38-s − 1.28·39-s + 0.609·43-s + ⋯

Functional equation

Λ(s)=((216⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅582^{16} \cdot 5^{8}
Sign: 11
Analytic conductor: 104.075104.075
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.59314470070.5931447007
L(12)L(\frac12) ≈\approx 0.59314470070.5931447007
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)
bad2C2C_2 (1+pT+pT2)2 ( 1 + p T + p T^{2} )^{2}
5 1 1
good3C2C_2×\timesC22C_2^2 (1+T+pT2)2(1−5T2+p2T4) ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} )
7D4×C2D_4\times C_2 1−4T2+58T4−4p2T6+p4T8 1 - 4 T^{2} + 58 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−6T+18T2−60T3+199T4−60pT5+18p2T6−6p3T7+p4T8 1 - 6 T + 18 T^{2} - 60 T^{3} + 199 T^{4} - 60 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−4T+8T2+28T3−302T4+28pT5+8p2T6−4p3T7+p4T8 1 - 4 T + 8 T^{2} + 28 T^{3} - 302 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
17D4D_{4} (1−4T+27T2−4pT3+p2T4)2 ( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−6T+18T2−108T3+647T4−108pT5+18p2T6−6p3T7+p4T8 1 - 6 T + 18 T^{2} - 108 T^{3} + 647 T^{4} - 108 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−52T2+1338T4−52p2T6+p4T8 1 - 52 T^{2} + 1338 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−4T+8T2−4pT3+p2T4)2 ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
31D4D_{4} (1−2T+52T2−2pT3+p2T4)2 ( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1+16T+128T2+752T3+4318T4+752pT5+128p2T6+16p3T7+p4T8 1 + 16 T + 128 T^{2} + 752 T^{3} + 4318 T^{4} + 752 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
41C22C_2^2 (1−57T2+p2T4)2 ( 1 - 57 T^{2} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−4T+8T2+4pT3−2p2T4+4p2T5+8p2T6−4p3T7+p4T8 1 - 4 T + 8 T^{2} + 4 p T^{3} - 2 p^{2} T^{4} + 4 p^{2} T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
53C23C_2^3 1+1438T4+p4T8 1 + 1438 T^{4} + p^{4} T^{8}
59D4×C2D_4\times C_2 1−8T+32T2−360T3+3854T4−360pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 360 T^{3} + 3854 T^{4} - 360 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−12T+72T2+108T3−4738T4+108pT5+72p2T6−12p3T7+p4T8 1 - 12 T + 72 T^{2} + 108 T^{3} - 4738 T^{4} + 108 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+30T+450T2+5220T3+49103T4+5220pT5+450p2T6+30p3T7+p4T8 1 + 30 T + 450 T^{2} + 5220 T^{3} + 49103 T^{4} + 5220 p T^{5} + 450 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−188T2+18214T4−188p2T6+p4T8 1 - 188 T^{2} + 18214 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8}
73D4×C2D_4\times C_2 1−70T2+7483T4−70p2T6+p4T8 1 - 70 T^{2} + 7483 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1−2T+148T2−2pT3+p2T4)2 ( 1 - 2 T + 148 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−22T+242T2−3036T3+35063T4−3036pT5+242p2T6−22p3T7+p4T8 1 - 22 T + 242 T^{2} - 3036 T^{3} + 35063 T^{4} - 3036 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}
89D4×C2D_4\times C_2 1−86T2+3435T4−86p2T6+p4T8 1 - 86 T^{2} + 3435 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8}
97C22C_2^2 (1+150T2+p2T4)2 ( 1 + 150 T^{2} + p^{2} T^{4} )^{2}
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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

−8.379962157056382291436883757890, −7.74688146716074226541689682474, −7.73760545097658337390851717339, −7.64410973305022098086438252038, −7.24254328040066159019561178917, −7.07412179564542852607016486749, −6.75373015284628119856514899684, −6.52864255387290912487410046196, −6.47362833642957498703992632024, −5.84955239770017212112471238498, −5.78441480650827764211035425254, −5.64290144240409760875496240929, −5.07635353255458939204698659288, −4.78067929007260748573436189825, −4.72069258549140689406239649492, −3.94836274476933997587892634598, −3.90692434001507114955212768179, −3.73068976830149818420888343179, −3.00081411423370612210622339578, −2.83111584436372240333552056504, −2.16761262225022100877252567896, −1.73709995370689421958360416190, −1.02974067913352657908585656083, −0.984444985671682020464278245201, −0.906736594402918891811455148430, 0.906736594402918891811455148430, 0.984444985671682020464278245201, 1.02974067913352657908585656083, 1.73709995370689421958360416190, 2.16761262225022100877252567896, 2.83111584436372240333552056504, 3.00081411423370612210622339578, 3.73068976830149818420888343179, 3.90692434001507114955212768179, 3.94836274476933997587892634598, 4.72069258549140689406239649492, 4.78067929007260748573436189825, 5.07635353255458939204698659288, 5.64290144240409760875496240929, 5.78441480650827764211035425254, 5.84955239770017212112471238498, 6.47362833642957498703992632024, 6.52864255387290912487410046196, 6.75373015284628119856514899684, 7.07412179564542852607016486749, 7.24254328040066159019561178917, 7.64410973305022098086438252038, 7.73760545097658337390851717339, 7.74688146716074226541689682474, 8.379962157056382291436883757890

Graph of the ZZ-function along the critical line