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Properties

Label 8-2160e4-1.1-c1e4-0-12
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
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  − 3·5-s + 6·7-s + 6·11-s + 6·17-s + 5·25-s − 18·35-s − 24·43-s + 11·49-s − 12·53-s − 18·55-s + 12·59-s + 22·61-s − 36·67-s + 36·77-s − 18·85-s + 22·109-s − 12·113-s + 36·119-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1.34·5-s + 2.26·7-s + 1.80·11-s + 1.45·17-s + 25-s − 3.04·35-s − 3.65·43-s + 11/7·49-s − 1.64·53-s − 2.42·55-s + 1.56·59-s + 2.81·61-s − 4.39·67-s + 4.10·77-s − 1.95·85-s + 2.10·109-s − 1.12·113-s + 3.30·119-s − 0.454·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \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: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 4.3588958704.358895870
L(12)L(\frac12) ≈\approx 4.3588958704.358895870
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
3 1 1
5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
good7D4D_{4} (1−3T+8T2−3pT3+p2T4)2 ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
11D4D_{4} (1−3T+16T2−3pT3+p2T4)2 ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−14T2+p2T4)2 ( 1 - 14 T^{2} + p^{2} T^{4} )^{2}
17D4D_{4} (1−3T+28T2−3pT3+p2T4)2 ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−25T2+804T4−25p2T6+p4T8 1 - 25 T^{2} + 804 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−41T2+1404T4−41p2T6+p4T8 1 - 41 T^{2} + 1404 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−40T2+894T4−40p2T6+p4T8 1 - 40 T^{2} + 894 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}
31C2C_2 (1−11T+pT2)2(1+11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)2(1+10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}
41D4×C2D_4\times C_2 1−88T2+4110T4−88p2T6+p4T8 1 - 88 T^{2} + 4110 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8}
43C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
47C22C_2^2 (1−82T2+p2T4)2 ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
59D4D_{4} (1−6T+94T2−6pT3+p2T4)2 ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1−11T+78T2−11pT3+p2T4)2 ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2}
67D4D_{4} (1+18T+182T2+18pT3+p2T4)2 ( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1+155T2+15996T4+155p2T6+p4T8 1 + 155 T^{2} + 15996 T^{4} + 155 p^{2} T^{6} + p^{4} T^{8}
79D4×C2D_4\times C_2 1−253T2+27816T4−253p2T6+p4T8 1 - 253 T^{2} + 27816 T^{4} - 253 p^{2} T^{6} + p^{4} T^{8}
83D4×C2D_4\times C_2 1−110T2+12051T4−110p2T6+p4T8 1 - 110 T^{2} + 12051 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8}
89C22C_2^2 (1−134T2+p2T4)2 ( 1 - 134 T^{2} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−217T2+24576T4−217p2T6+p4T8 1 - 217 T^{2} + 24576 T^{4} - 217 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

−6.33660322551234355351786961849, −6.25384774509474832836242725545, −6.19068819405059673960759778215, −5.61673595894994280939766238299, −5.56034292085955261996339090407, −5.53454965989862479457796541126, −4.95010914475644588452577792079, −4.89336054713826320562887304130, −4.81260101018612622303825195375, −4.53331293587707406749530955766, −4.37701047840381230615854579196, −4.13094447126929714585926569703, −3.81195323282954072651855807899, −3.62118755559918085025717421857, −3.38086182354188254117451582120, −3.32813640911016778759035261535, −2.90931706848926145515413634493, −2.78220779954210796095913006398, −2.14847542899726284688530348918, −1.87534219317307335857125956011, −1.64183104121081079248956001210, −1.55856239782758208204073208414, −1.18466151783060763859128630815, −0.811966546306140975447677245221, −0.37613737841077933550109035533, 0.37613737841077933550109035533, 0.811966546306140975447677245221, 1.18466151783060763859128630815, 1.55856239782758208204073208414, 1.64183104121081079248956001210, 1.87534219317307335857125956011, 2.14847542899726284688530348918, 2.78220779954210796095913006398, 2.90931706848926145515413634493, 3.32813640911016778759035261535, 3.38086182354188254117451582120, 3.62118755559918085025717421857, 3.81195323282954072651855807899, 4.13094447126929714585926569703, 4.37701047840381230615854579196, 4.53331293587707406749530955766, 4.81260101018612622303825195375, 4.89336054713826320562887304130, 4.95010914475644588452577792079, 5.53454965989862479457796541126, 5.56034292085955261996339090407, 5.61673595894994280939766238299, 6.19068819405059673960759778215, 6.25384774509474832836242725545, 6.33660322551234355351786961849

Graph of the ZZ-function along the critical line