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Properties

Label 8-294e4-1.1-c3e4-0-1
Degree 88
Conductor 74711820967471182096
Sign 11
Analytic cond. 90542.790542.7
Root an. cond. 4.164924.16492
Motivic weight 33
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  − 4·2-s + 6·3-s + 4·4-s + 5·5-s − 24·6-s + 16·8-s + 9·9-s − 20·10-s − 67·11-s + 24·12-s − 82·13-s + 30·15-s − 64·16-s − 92·17-s − 36·18-s + 43·19-s + 20·20-s + 268·22-s − 148·23-s + 96·24-s − 80·25-s + 328·26-s − 54·27-s + 154·29-s − 120·30-s − 520·31-s + 64·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 0.632·10-s − 1.83·11-s + 0.577·12-s − 1.74·13-s + 0.516·15-s − 16-s − 1.31·17-s − 0.471·18-s + 0.519·19-s + 0.223·20-s + 2.59·22-s − 1.34·23-s + 0.816·24-s − 0.639·25-s + 2.47·26-s − 0.384·27-s + 0.986·29-s − 0.730·30-s − 3.01·31-s + 0.353·32-s + ⋯

Functional equation

Λ(s)=((24⋅34⋅78)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((24⋅34⋅78)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅34⋅782^{4} \cdot 3^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 90542.790542.7
Root analytic conductor: 4.164924.16492
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅34⋅78, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.091305210290.09130521029
L(12)L(\frac12) ≈\approx 0.091305210290.09130521029
L(52)L(\frac{5}{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+p2T2)2 ( 1 + p T + p^{2} T^{2} )^{2}
3C2C_2 (1−pT+p2T2)2 ( 1 - p T + p^{2} T^{2} )^{2}
7 1 1
good5D4×C2D_4\times C_2 1−pT+21pT2+66p2T3−494p2T4+66p5T5+21p7T6−p10T7+p12T8 1 - p T + 21 p T^{2} + 66 p^{2} T^{3} - 494 p^{2} T^{4} + 66 p^{5} T^{5} + 21 p^{7} T^{6} - p^{10} T^{7} + p^{12} T^{8}
11D4×C2D_4\times C_2 1+67T+1041T2+52662T3+4142284T4+52662p3T5+1041p6T6+67p9T7+p12T8 1 + 67 T + 1041 T^{2} + 52662 T^{3} + 4142284 T^{4} + 52662 p^{3} T^{5} + 1041 p^{6} T^{6} + 67 p^{9} T^{7} + p^{12} T^{8}
13D4D_{4} (1+41T+4478T2+41p3T3+p6T4)2 ( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} )^{2}
17D4×C2D_4\times C_2 1+92T+1902T2−17664pT3−78413p2T4−17664p4T5+1902p6T6+92p9T7+p12T8 1 + 92 T + 1902 T^{2} - 17664 p T^{3} - 78413 p^{2} T^{4} - 17664 p^{4} T^{5} + 1902 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8}
19D4×C2D_4\times C_2 1−43T−9305T2+110252T3+64683544T4+110252p3T5−9305p6T6−43p9T7+p12T8 1 - 43 T - 9305 T^{2} + 110252 T^{3} + 64683544 T^{4} + 110252 p^{3} T^{5} - 9305 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8}
23D4×C2D_4\times C_2 1+148T−2526T2+14208T3+182283043T4+14208p3T5−2526p6T6+148p9T7+p12T8 1 + 148 T - 2526 T^{2} + 14208 T^{3} + 182283043 T^{4} + 14208 p^{3} T^{5} - 2526 p^{6} T^{6} + 148 p^{9} T^{7} + p^{12} T^{8}
29D4D_{4} (1−77T+9574T2−77p3T3+p6T4)2 ( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} )^{2}
31D4×C2D_4\times C_2 1+520T+144563T2+34452600T3+6891960488T4+34452600p3T5+144563p6T6+520p9T7+p12T8 1 + 520 T + 144563 T^{2} + 34452600 T^{3} + 6891960488 T^{4} + 34452600 p^{3} T^{5} + 144563 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8}
37D4×C2D_4\times C_2 1+7T−74033T2−190568T3+2919934318T4−190568p3T5−74033p6T6+7p9T7+p12T8 1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 190568 p^{3} T^{5} - 74033 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8}
41D4D_{4} (1−426T+171106T2−426p3T3+p6T4)2 ( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} )^{2}
43D4D_{4} (1+107T+86220T2+107p3T3+p6T4)2 ( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} )^{2}
47D4×C2D_4\times C_2 1−576T+89606T2−19885824T3+13421113923T4−19885824p3T5+89606p6T6−576p9T7+p12T8 1 - 576 T + 89606 T^{2} - 19885824 T^{3} + 13421113923 T^{4} - 19885824 p^{3} T^{5} + 89606 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8}
53D4×C2D_4\times C_2 1−243T−250441T2−2851848T3+64828660998T4−2851848p3T5−250441p6T6−243p9T7+p12T8 1 - 243 T - 250441 T^{2} - 2851848 T^{3} + 64828660998 T^{4} - 2851848 p^{3} T^{5} - 250441 p^{6} T^{6} - 243 p^{9} T^{7} + p^{12} T^{8}
59D4×C2D_4\times C_2 1+7T−200565T2−1471008T3−1944620216T4−1471008p3T5−200565p6T6+7p9T7+p12T8 1 + 7 T - 200565 T^{2} - 1471008 T^{3} - 1944620216 T^{4} - 1471008 p^{3} T^{5} - 200565 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8}
61D4×C2D_4\times C_2 1−224T−410950T2−1604736T3+149727814859T4−1604736p3T5−410950p6T6−224p9T7+p12T8 1 - 224 T - 410950 T^{2} - 1604736 T^{3} + 149727814859 T^{4} - 1604736 p^{3} T^{5} - 410950 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8}
67D4×C2D_4\times C_2 1+687T−206863T2+53109222T3+228403689708T4+53109222p3T5−206863p6T6+687p9T7+p12T8 1 + 687 T - 206863 T^{2} + 53109222 T^{3} + 228403689708 T^{4} + 53109222 p^{3} T^{5} - 206863 p^{6} T^{6} + 687 p^{9} T^{7} + p^{12} T^{8}
71D4D_{4} (1−472T+637018T2−472p3T3+p6T4)2 ( 1 - 472 T + 637018 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} )^{2}
73D4×C2D_4\times C_2 1+921T+270053T2−184058166T3−147013032042T4−184058166p3T5+270053p6T6+921p9T7+p12T8 1 + 921 T + 270053 T^{2} - 184058166 T^{3} - 147013032042 T^{4} - 184058166 p^{3} T^{5} + 270053 p^{6} T^{6} + 921 p^{9} T^{7} + p^{12} T^{8}
79D4×C2D_4\times C_2 1−526T−757051T2−25063374T3+689091996644T4−25063374p3T5−757051p6T6−526p9T7+p12T8 1 - 526 T - 757051 T^{2} - 25063374 T^{3} + 689091996644 T^{4} - 25063374 p^{3} T^{5} - 757051 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8}
83D4D_{4} (1−221T+945628T2−221p3T3+p6T4)2 ( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} )^{2}
89D4×C2D_4\times C_2 1−774T−367486T2+343173024T3+14930800239T4+343173024p3T5−367486p6T6−774p9T7+p12T8 1 - 774 T - 367486 T^{2} + 343173024 T^{3} + 14930800239 T^{4} + 343173024 p^{3} T^{5} - 367486 p^{6} T^{6} - 774 p^{9} T^{7} + p^{12} T^{8}
97D4D_{4} (1+1953T+2366992T2+1953p3T3+p6T4)2 ( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} 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

−8.018501530828628551448918060774, −7.936480338683790001176559096966, −7.55565003644032229753366525302, −7.52479019802730056949740865507, −7.46122620358906704236580464815, −7.00465529752423820732720255064, −6.76791457889360561633335853008, −6.33072367385129767772116084212, −5.93609403693294786822780329581, −5.78023253182408109812016082003, −5.20767890743804762396596558351, −5.20122457338249944588479538557, −5.12771536620029890485343004511, −4.31237766489417542031878253864, −4.25416374499708619914777383916, −3.83659349463606102453932133079, −3.74041758720168146754045239081, −2.72249058556384181380223808557, −2.70281736935727476984077404219, −2.60103290262005161987481281344, −2.16788358256481329236877733480, −1.82244994494784831939437671044, −1.34335788745930428083131141337, −0.55162803863153112264212906285, −0.089878875687861273206985580797, 0.089878875687861273206985580797, 0.55162803863153112264212906285, 1.34335788745930428083131141337, 1.82244994494784831939437671044, 2.16788358256481329236877733480, 2.60103290262005161987481281344, 2.70281736935727476984077404219, 2.72249058556384181380223808557, 3.74041758720168146754045239081, 3.83659349463606102453932133079, 4.25416374499708619914777383916, 4.31237766489417542031878253864, 5.12771536620029890485343004511, 5.20122457338249944588479538557, 5.20767890743804762396596558351, 5.78023253182408109812016082003, 5.93609403693294786822780329581, 6.33072367385129767772116084212, 6.76791457889360561633335853008, 7.00465529752423820732720255064, 7.46122620358906704236580464815, 7.52479019802730056949740865507, 7.55565003644032229753366525302, 7.936480338683790001176559096966, 8.018501530828628551448918060774

Graph of the ZZ-function along the critical line