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Properties

Label 8-55e4-1.1-c1e4-0-2
Degree 88
Conductor 91506259150625
Sign 11
Analytic cond. 0.03720130.0372013
Root an. cond. 0.6627040.662704
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-s − 3·5-s + 5·9-s − 4·11-s + 16-s − 16·19-s − 3·20-s + 5·25-s − 12·29-s + 2·31-s + 5·36-s + 12·41-s − 4·44-s − 15·45-s + 4·49-s + 12·55-s + 18·59-s + 20·61-s + 5·64-s − 6·71-s − 16·76-s − 28·79-s − 3·80-s + 9·81-s + 6·89-s + 48·95-s − 20·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.34·5-s + 5/3·9-s − 1.20·11-s + 1/4·16-s − 3.67·19-s − 0.670·20-s + 25-s − 2.22·29-s + 0.359·31-s + 5/6·36-s + 1.87·41-s − 0.603·44-s − 2.23·45-s + 4/7·49-s + 1.61·55-s + 2.34·59-s + 2.56·61-s + 5/8·64-s − 0.712·71-s − 1.83·76-s − 3.15·79-s − 0.335·80-s + 81-s + 0.635·89-s + 4.92·95-s − 2.01·99-s + ⋯

Functional equation

Λ(s)=(9150625s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9150625s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 91506259150625    =    54⋅1145^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 0.03720130.0372013
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 9150625, ( :1/2,1/2,1/2,1/2), 1)(8,\ 9150625,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.49565204530.4956520453
L(12)L(\frac12) ≈\approx 0.49565204530.4956520453
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)
bad5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C1C_1 (1+T)4 ( 1 + T )^{4}
good2D4×C2D_4\times C_2 1−T2−p2T6+p4T8 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8}
3C22C_2^2×\timesC22C_2^2 (1−T−2T2−pT3+p2T4)(1+T−2T2+pT3+p2T4) ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )
7C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17D4×C2D_4\times C_2 1−40T2+846T4−40p2T6+p4T8 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}
19C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
23D4×C2D_4\times C_2 1−85T2+2856T4−85p2T6+p4T8 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8}
29D4D_{4} (1+6T+34T2+6pT3+p2T4)2 ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
31D4D_{4} (1−T+54T2−pT3+p2T4)2 ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2}
37C22C_2^2×\timesC22C_2^2 (1−7T+12T2−7pT3+p2T4)(1+7T+12T2+7pT3+p2T4) ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )
41D4D_{4} (1−6T+58T2−6pT3+p2T4)2 ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−74T2+p2T4)2 ( 1 - 74 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
53D4×C2D_4\times C_2 1−100T2+6006T4−100p2T6+p4T8 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1−9T+130T2−9pT3+p2T4)2 ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1−10T+114T2−10pT3+p2T4)2 ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−181T2+15312T4−181p2T6+p4T8 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1+3T+70T2+3pT3+p2T4)2 ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−98T2+p2T4)2 ( 1 - 98 T^{2} + p^{2} T^{4} )^{2}
79D4D_{4} (1+14T+174T2+14pT3+p2T4)2 ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−122T2+p2T4)2 ( 1 - 122 T^{2} + p^{2} T^{4} )^{2}
89D4D_{4} (1−3T+172T2−3pT3+p2T4)2 ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−337T2+47136T4−337p2T6+p4T8 1 - 337 T^{2} + 47136 T^{4} - 337 p^{2} T^{6} + 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

−11.51323317728160153252196799477, −11.23905208913192929239537573227, −10.76708718026809165163786206272, −10.55003680888481659010168730285, −10.34868265170857292905220817206, −10.21784533693697580003851458916, −9.572150857601680511869988955596, −9.384390295239582833487040197405, −8.752358483478454298537765325143, −8.561913929967990611253256812393, −8.261424559393711865296506807208, −7.75464132179680785468636652988, −7.72407164836328248452341648958, −7.16134975081704456913463737652, −6.85437210822826851515912330194, −6.76134604263451556055217846691, −6.13306182954689398397886189556, −5.52923801730463138407263999870, −5.42054206796080839394948654022, −4.38975127802683186965272439597, −4.27157937693817625799324775125, −4.12586615836761186344194460914, −3.47984059806640897081261858555, −2.45217711766523776713592093680, −2.09429720371933036108397787319, 2.09429720371933036108397787319, 2.45217711766523776713592093680, 3.47984059806640897081261858555, 4.12586615836761186344194460914, 4.27157937693817625799324775125, 4.38975127802683186965272439597, 5.42054206796080839394948654022, 5.52923801730463138407263999870, 6.13306182954689398397886189556, 6.76134604263451556055217846691, 6.85437210822826851515912330194, 7.16134975081704456913463737652, 7.72407164836328248452341648958, 7.75464132179680785468636652988, 8.261424559393711865296506807208, 8.561913929967990611253256812393, 8.752358483478454298537765325143, 9.384390295239582833487040197405, 9.572150857601680511869988955596, 10.21784533693697580003851458916, 10.34868265170857292905220817206, 10.55003680888481659010168730285, 10.76708718026809165163786206272, 11.23905208913192929239537573227, 11.51323317728160153252196799477

Graph of the ZZ-function along the critical line