↑

Properties

Label 8-336e4-1.1-c2e4-0-9
Degree 88
Conductor 1274550681612745506816
Sign 11
Analytic cond. 7025.827025.82
Root an. cond. 3.025773.02577
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 12·5-s + 10·7-s + 21·9-s + 12·11-s − 72·15-s − 48·17-s + 42·19-s − 60·21-s − 24·23-s + 58·25-s − 54·27-s − 102·31-s − 72·33-s + 120·35-s + 22·37-s − 28·43-s + 252·45-s + 132·47-s + 49·49-s + 288·51-s + 120·53-s + 144·55-s − 252·57-s + 24·59-s − 72·61-s + 210·63-s + ⋯
L(s)  = 1  − 2·3-s + 12/5·5-s + 10/7·7-s + 7/3·9-s + 1.09·11-s − 4.79·15-s − 2.82·17-s + 2.21·19-s − 2.85·21-s − 1.04·23-s + 2.31·25-s − 2·27-s − 3.29·31-s − 2.18·33-s + 24/7·35-s + 0.594·37-s − 0.651·43-s + 28/5·45-s + 2.80·47-s + 49-s + 5.64·51-s + 2.26·53-s + 2.61·55-s − 4.42·57-s + 0.406·59-s − 1.18·61-s + 10/3·63-s + ⋯

Functional equation

Λ(s)=((216⋅34⋅74)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅34⋅74)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅34⋅742^{16} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 7025.827025.82
Root analytic conductor: 3.025773.02577
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅34⋅74, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.5985510511.598551051
L(12)L(\frac12) ≈\approx 1.5985510511.598551051
L(2)L(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
3C2C_2 (1+pT+pT2)2 ( 1 + p T + p T^{2} )^{2}
7C22C_2^2 1−10T+51T2−10p2T3+p4T4 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4}
good5D4×C2D_4\times C_2 1−12T+86T2−456T3+2019T4−456p2T5+86p4T6−12p6T7+p8T8 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 456 p^{2} T^{5} + 86 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8}
11C22C_2^2 (1−6T−85T2−6p2T3+p4T4)2 ( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2}
13D4×C2D_4\times C_2 1+98T2+54915T4+98p4T6+p8T8 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8}
17D4×C2D_4\times C_2 1+48T+1514T2+35808T3+694947T4+35808p2T5+1514p4T6+48p6T7+p8T8 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 35808 p^{2} T^{5} + 1514 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8}
19D4×C2D_4\times C_2 1−42T+1433T2−35490T3+795972T4−35490p2T5+1433p4T6−42p6T7+p8T8 1 - 42 T + 1433 T^{2} - 35490 T^{3} + 795972 T^{4} - 35490 p^{2} T^{5} + 1433 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8}
23D4×C2D_4\times C_2 1+24T+22T2−12096T3−277629T4−12096p2T5+22p4T6+24p6T7+p8T8 1 + 24 T + 22 T^{2} - 12096 T^{3} - 277629 T^{4} - 12096 p^{2} T^{5} + 22 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8}
29C22C_2^2 (1+530T2+p4T4)2 ( 1 + 530 T^{2} + p^{4} T^{4} )^{2}
31D4×C2D_4\times C_2 1+102T+6041T2+8466pT3+9396p2T4+8466p3T5+6041p4T6+102p6T7+p8T8 1 + 102 T + 6041 T^{2} + 8466 p T^{3} + 9396 p^{2} T^{4} + 8466 p^{3} T^{5} + 6041 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8}
37D4×C2D_4\times C_2 1−22T−2087T2+3674T3+4073284T4+3674p2T5−2087p4T6−22p6T7+p8T8 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8}
41D4×C2D_4\times C_2 1−2476T2+6405414T4−2476p4T6+p8T8 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1+14T+3675T2+14p2T3+p4T4)2 ( 1 + 14 T + 3675 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−132T+11654T2−771672T3+42125907T4−771672p2T5+11654p4T6−132p6T7+p8T8 1 - 132 T + 11654 T^{2} - 771672 T^{3} + 42125907 T^{4} - 771672 p^{2} T^{5} + 11654 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−120T+110pT2−354240T3+25104819T4−354240p2T5+110p5T6−120p6T7+p8T8 1 - 120 T + 110 p T^{2} - 354240 T^{3} + 25104819 T^{4} - 354240 p^{2} T^{5} + 110 p^{5} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−24T+6026T2−140016T3+22586547T4−140016p2T5+6026p4T6−24p6T7+p8T8 1 - 24 T + 6026 T^{2} - 140016 T^{3} + 22586547 T^{4} - 140016 p^{2} T^{5} + 6026 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8}
61D4×C2D_4\times C_2 1+72T+9218T2+539280T3+48684147T4+539280p2T5+9218p4T6+72p6T7+p8T8 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8}
67D4×C2D_4\times C_2 1+110T+3625T2−55330T3−2642396T4−55330p2T5+3625p4T6+110p6T7+p8T8 1 + 110 T + 3625 T^{2} - 55330 T^{3} - 2642396 T^{4} - 55330 p^{2} T^{5} + 3625 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8}
71D4D_{4} (1+156T+178pT2+156p2T3+p4T4)2 ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2}
73D4×C2D_4\times C_2 1+66T+2873T2+93786T3−18641292T4+93786p2T5+2873p4T6+66p6T7+p8T8 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 93786 p^{2} T^{5} + 2873 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8}
79D4×C2D_4\times C_2 1−10T−3695T2+86870T3−25172156T4+86870p2T5−3695p4T6−10p6T7+p8T8 1 - 10 T - 3695 T^{2} + 86870 T^{3} - 25172156 T^{4} + 86870 p^{2} T^{5} - 3695 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−116pT2+107694438T4−116p5T6+p8T8 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8}
89C22C_2^2 (1+36T+8353T2+36p2T3+p4T4)2 ( 1 + 36 T + 8353 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4×C2D_4\times C_2 1−36580T2+511416774T4−36580p4T6+p8T8 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8}
show more
show less
   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

−8.151835481077152472907957382424, −7.65419972440289389774311525835, −7.48418616250043892695894158533, −7.25289639055717817922697493637, −7.14497368805639321203491959667, −6.90785231855669327180410414564, −6.42123671748674941584836250427, −6.22123382742639049479108146886, −5.92285178242963670527150351303, −5.83396232702134959754649287498, −5.59350061630147971114709242051, −5.35241684172126701457131833665, −5.29085586208094683692011100207, −4.58708171728656882885678640825, −4.54094877279726719753516454754, −4.39094462496383554117165315713, −3.82836960282743377088542992836, −3.68878024531006319202674696792, −2.89814070708283217346319003471, −2.35070845570825123393360283764, −2.24475340385583045637862320816, −1.65233983664443142631883754336, −1.49375913020863040215864666091, −1.23213891163193174025746937530, −0.30694248599785445817418134248, 0.30694248599785445817418134248, 1.23213891163193174025746937530, 1.49375913020863040215864666091, 1.65233983664443142631883754336, 2.24475340385583045637862320816, 2.35070845570825123393360283764, 2.89814070708283217346319003471, 3.68878024531006319202674696792, 3.82836960282743377088542992836, 4.39094462496383554117165315713, 4.54094877279726719753516454754, 4.58708171728656882885678640825, 5.29085586208094683692011100207, 5.35241684172126701457131833665, 5.59350061630147971114709242051, 5.83396232702134959754649287498, 5.92285178242963670527150351303, 6.22123382742639049479108146886, 6.42123671748674941584836250427, 6.90785231855669327180410414564, 7.14497368805639321203491959667, 7.25289639055717817922697493637, 7.48418616250043892695894158533, 7.65419972440289389774311525835, 8.151835481077152472907957382424

Graph of the ZZ-function along the critical line