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Properties

Label 8-960e4-1.1-c1e4-0-17
Degree 88
Conductor 849346560000849346560000
Sign 11
Analytic cond. 3452.973452.97
Root an. cond. 2.768682.76868
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 8·9-s + 8·13-s − 24·23-s − 2·25-s + 12·27-s + 24·37-s + 32·39-s + 8·47-s + 16·49-s − 16·59-s − 96·69-s − 16·71-s + 40·73-s − 8·75-s + 23·81-s − 8·83-s + 8·97-s − 56·107-s − 48·109-s + 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2.30·3-s + 8/3·9-s + 2.21·13-s − 5.00·23-s − 2/5·25-s + 2.30·27-s + 3.94·37-s + 5.12·39-s + 1.16·47-s + 16/7·49-s − 2.08·59-s − 11.5·69-s − 1.89·71-s + 4.68·73-s − 0.923·75-s + 23/9·81-s − 0.878·83-s + 0.812·97-s − 5.41·107-s − 4.59·109-s + 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=((224⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅34⋅542^{24} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 3452.973452.97
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.8361049506.836104950
L(12)L(\frac12) ≈\approx 6.8361049506.836104950
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)
bad2 1 1
3C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−16T2+130T4−16p2T6+p4T8 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
17C4×C2C_4\times C_2 1+4T2+70T4+4p2T6+p4T8 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
23D4D_{4} (1+12T+80T2+12pT3+p2T4)2 ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
37D4D_{4} (1−12T+78T2−12pT3+p2T4)2 ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
41C22C_2^2 (1−78T2+p2T4)2 ( 1 - 78 T^{2} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+32T2+2386T4+32p2T6+p4T8 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1−4T+80T2−4pT3+p2T4)2 ( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−140T2+10006T4−140p2T6+p4T8 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+8T+102T2+8pT3+p2T4)2 ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+90T2+p2T4)2 ( 1 + 90 T^{2} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−96T2+4082T4−96p2T6+p4T8 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1+8T+150T2+8pT3+p2T4)2 ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
73C4C_4 (1−20T+214T2−20pT3+p2T4)2 ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−140T2+12774T4−140p2T6+p4T8 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
83D4D_{4} (1+4T+120T2+4pT3+p2T4)2 ( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1−68T2+8806T4−68p2T6+p4T8 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}
97C4C_4 (1−4T−90T2−4pT3+p2T4)2 ( 1 - 4 T - 90 T^{2} - 4 p T^{3} + 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

−7.67791625374048439321451067749, −6.73000673676486891701030068145, −6.70358510862911210088224607376, −6.68616512729359268528128417691, −6.26216519437737281059893779057, −6.00001014784392624777439472446, −5.97051709847828387592251343797, −5.71450596088105319919339095239, −5.33476764644705104083583278791, −5.30510685353685002044739947340, −4.49469931838036930045635179805, −4.38368658183573462338288319609, −4.27088844762796763341626304012, −3.90962579124419425007846334516, −3.74522112318939901511423993579, −3.66182829044067056772071800503, −3.59988102841603146407170485291, −2.66388554224525805395656689073, −2.60689882874429319990856351568, −2.51780175269678124865062626304, −2.45284117061914577248896877768, −1.73612211265992309580400064701, −1.43274033432367894065856734640, −1.28921250116194665075053671117, −0.47026160017617460907234407445, 0.47026160017617460907234407445, 1.28921250116194665075053671117, 1.43274033432367894065856734640, 1.73612211265992309580400064701, 2.45284117061914577248896877768, 2.51780175269678124865062626304, 2.60689882874429319990856351568, 2.66388554224525805395656689073, 3.59988102841603146407170485291, 3.66182829044067056772071800503, 3.74522112318939901511423993579, 3.90962579124419425007846334516, 4.27088844762796763341626304012, 4.38368658183573462338288319609, 4.49469931838036930045635179805, 5.30510685353685002044739947340, 5.33476764644705104083583278791, 5.71450596088105319919339095239, 5.97051709847828387592251343797, 6.00001014784392624777439472446, 6.26216519437737281059893779057, 6.68616512729359268528128417691, 6.70358510862911210088224607376, 6.73000673676486891701030068145, 7.67791625374048439321451067749

Graph of the ZZ-function along the critical line