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Properties

Label 8-2160e4-1.1-c1e4-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 3·11-s + 2·13-s + 18·17-s − 2·19-s + 3·23-s + 25-s − 3·29-s − 2·31-s − 2·35-s − 16·37-s − 6·41-s − 17·43-s + 9·47-s + 6·49-s − 6·55-s − 3·59-s − 61-s − 4·65-s − 14·67-s − 24·71-s − 22·73-s + 3·77-s + 4·79-s − 9·83-s − 36·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s + 4.36·17-s − 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s − 0.338·35-s − 2.63·37-s − 0.937·41-s − 2.59·43-s + 1.31·47-s + 6/7·49-s − 0.809·55-s − 0.390·59-s − 0.128·61-s − 0.496·65-s − 1.71·67-s − 2.84·71-s − 2.57·73-s + 0.341·77-s + 0.450·79-s − 0.987·83-s − 3.90·85-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 0.39131650710.3913165071
L(12)L(\frac12) ≈\approx 0.39131650710.3913165071
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
5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−T−5T2+8T3−20T4+8pT5−5p2T6−p3T7+p4T8 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−3T−7T2+18T3+36T4+18pT5−7p2T6−3p3T7+p4T8 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−2T+10T2+64T3−185T4+64pT5+10p2T6−2p3T7+p4T8 1 - 2 T + 10 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
17D4D_{4} (1−9T+46T2−9pT3+p2T4)2 ( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
19D4D_{4} (1+T+30T2+pT3+p2T4)2 ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−3T−31T2+18T3+864T4+18pT5−31p2T6−3p3T7+p4T8 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 18 p T^{5} - 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+3T−43T2−18T3+1602T4−18pT5−43p2T6+3p3T7+p4T8 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1+2T−26T2−64T3−185T4−64pT5−26p2T6+2p3T7+p4T8 1 + 2 T - 26 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
37C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
41C22C_2^2 (1+3T−32T2+3pT3+p2T4)2 ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+17T+139T2+1088T3+8224T4+1088pT5+139p2T6+17p3T7+p4T8 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−9T−25T2−108T3+5220T4−108pT5−25p2T6−9p3T7+p4T8 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
53C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+3T−103T2−18T3+8532T4−18pT5−103p2T6+3p3T7+p4T8 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 18 p T^{5} - 103 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+T−47T2−74T3−1478T4−74pT5−47p2T6+p3T7+p4T8 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
67C22C_2^2 (1+7T−18T2+7pT3+p2T4)2 ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
71C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
73D4D_{4} (1+11T+102T2+11pT3+p2T4)2 ( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−2T−75T2−2pT3+p2T4)2 ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1+9T−97T2+108T3+18072T4+108pT5−97p2T6+9p3T7+p4T8 1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 108 p T^{5} - 97 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1+15T+160T2+15pT3+p2T4)2 ( 1 + 15 T + 160 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−11T−95T2−242T3+25510T4−242pT5−95p2T6−11p3T7+p4T8 1 - 11 T - 95 T^{2} - 242 T^{3} + 25510 T^{4} - 242 p T^{5} - 95 p^{2} T^{6} - 11 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

−6.60719439416172647058980097680, −6.04861493998483332078206496686, −5.84445433871508404300560822672, −5.84315399154088439119580258964, −5.57894415115371518130931005317, −5.49055643837319231914326888234, −5.41579970014688605350626889843, −4.77664744588682740242015216761, −4.75876216426576077822711256841, −4.61386431427759819082598315660, −4.32545865104193855440291381480, −4.01660146687051783073530087133, −3.71856314478104067363303091762, −3.48297232477116297028425260396, −3.40561046180895867081244788786, −3.33554607798702019121595924936, −2.97106047633670901965720122850, −2.82065025277579673759799089896, −2.40724324228254774161873856449, −1.68980191857602202439598792166, −1.65818213202416626105002197996, −1.49284249398105826957327862181, −1.23443358669135368815778335351, −0.867548070377326672663479227381, −0.10535307833362608397612487114, 0.10535307833362608397612487114, 0.867548070377326672663479227381, 1.23443358669135368815778335351, 1.49284249398105826957327862181, 1.65818213202416626105002197996, 1.68980191857602202439598792166, 2.40724324228254774161873856449, 2.82065025277579673759799089896, 2.97106047633670901965720122850, 3.33554607798702019121595924936, 3.40561046180895867081244788786, 3.48297232477116297028425260396, 3.71856314478104067363303091762, 4.01660146687051783073530087133, 4.32545865104193855440291381480, 4.61386431427759819082598315660, 4.75876216426576077822711256841, 4.77664744588682740242015216761, 5.41579970014688605350626889843, 5.49055643837319231914326888234, 5.57894415115371518130931005317, 5.84315399154088439119580258964, 5.84445433871508404300560822672, 6.04861493998483332078206496686, 6.60719439416172647058980097680

Graph of the ZZ-function along the critical line