Properties

Label 8-384e4-1.1-c7e4-0-7
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.07055\times 10^{8}$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 108·3-s − 192·5-s − 680·7-s + 7.29e3·9-s − 4.49e3·11-s + 1.28e4·13-s − 2.07e4·15-s − 1.49e4·17-s − 2.15e4·19-s − 7.34e4·21-s − 5.49e4·23-s − 4.24e4·25-s + 3.93e5·27-s − 2.42e5·29-s − 1.51e5·31-s − 4.85e5·33-s + 1.30e5·35-s − 1.13e5·37-s + 1.38e6·39-s − 2.39e5·41-s + 1.49e6·43-s − 1.39e6·45-s − 7.72e5·47-s − 2.20e6·49-s − 1.61e6·51-s − 2.38e6·53-s + 8.63e5·55-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.686·5-s − 0.749·7-s + 10/3·9-s − 1.01·11-s + 1.62·13-s − 1.58·15-s − 0.738·17-s − 0.719·19-s − 1.73·21-s − 0.942·23-s − 0.543·25-s + 3.84·27-s − 1.84·29-s − 0.912·31-s − 2.35·33-s + 0.514·35-s − 0.367·37-s + 3.74·39-s − 0.541·41-s + 2.86·43-s − 2.28·45-s − 1.08·47-s − 2.67·49-s − 1.70·51-s − 2.20·53-s + 0.699·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.07055\times 10^{8}\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p^{3} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 192 T + 79316 T^{2} + 4433984 p T^{3} + 126774822 p^{2} T^{4} + 4433984 p^{8} T^{5} + 79316 p^{14} T^{6} + 192 p^{21} T^{7} + p^{28} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 680 T + 2666836 T^{2} + 1272950888 T^{3} + 2998744001542 T^{4} + 1272950888 p^{7} T^{5} + 2666836 p^{14} T^{6} + 680 p^{21} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4496 T + 26191340 T^{2} + 181969297168 T^{3} + 813341110124918 T^{4} + 181969297168 p^{7} T^{5} + 26191340 p^{14} T^{6} + 4496 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 12840 T + 132115948 T^{2} - 713027073272 T^{3} + 6498619482769302 T^{4} - 713027073272 p^{7} T^{5} + 132115948 p^{14} T^{6} - 12840 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 14952 T + 113915420 T^{2} + 5793309697112 T^{3} + 258267405954490374 T^{4} + 5793309697112 p^{7} T^{5} + 113915420 p^{14} T^{6} + 14952 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 21504 T + 2446733932 T^{2} + 53249972851712 T^{3} + 2887387256916153750 T^{4} + 53249972851712 p^{7} T^{5} + 2446733932 p^{14} T^{6} + 21504 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 54992 T + 1516634492 T^{2} - 249648798850544 T^{3} - 23815362010402645018 T^{4} - 249648798850544 p^{7} T^{5} + 1516634492 p^{14} T^{6} + 54992 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 242384 T + 34768900820 T^{2} + 146365951614192 p T^{3} + \)\(60\!\cdots\!98\)\( T^{4} + 146365951614192 p^{8} T^{5} + 34768900820 p^{14} T^{6} + 242384 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 151432 T + 93181101748 T^{2} + 9876646593730824 T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + 9876646593730824 p^{7} T^{5} + 93181101748 p^{14} T^{6} + 151432 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 113288 T + 266022576076 T^{2} + 28745603031036632 T^{3} + \)\(34\!\cdots\!18\)\( T^{4} + 28745603031036632 p^{7} T^{5} + 266022576076 p^{14} T^{6} + 113288 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 239176 T + 326935809404 T^{2} + 207369357289681464 T^{3} + \)\(56\!\cdots\!02\)\( T^{4} + 207369357289681464 p^{7} T^{5} + 326935809404 p^{14} T^{6} + 239176 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 1495328 T + 1369179794572 T^{2} - 952507717741307424 T^{3} + \)\(55\!\cdots\!42\)\( T^{4} - 952507717741307424 p^{7} T^{5} + 1369179794572 p^{14} T^{6} - 1495328 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 772368 T + 415239683612 T^{2} + 145238618306648272 T^{3} + \)\(28\!\cdots\!66\)\( T^{4} + 145238618306648272 p^{7} T^{5} + 415239683612 p^{14} T^{6} + 772368 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 2389776 T + 6682144612724 T^{2} + 9100205902823940720 T^{3} + \)\(13\!\cdots\!14\)\( T^{4} + 9100205902823940720 p^{7} T^{5} + 6682144612724 p^{14} T^{6} + 2389776 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 141232 T + 4417645030316 T^{2} - 4475890865907499312 T^{3} + \)\(94\!\cdots\!02\)\( T^{4} - 4475890865907499312 p^{7} T^{5} + 4417645030316 p^{14} T^{6} - 141232 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1231304 T + 7316522672620 T^{2} + 136484534453100856 p T^{3} + \)\(33\!\cdots\!14\)\( T^{4} + 136484534453100856 p^{8} T^{5} + 7316522672620 p^{14} T^{6} + 1231304 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 441392 T + 18348417993868 T^{2} + 6841337925338579248 T^{3} + \)\(15\!\cdots\!74\)\( T^{4} + 6841337925338579248 p^{7} T^{5} + 18348417993868 p^{14} T^{6} + 441392 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1507504 T + 18085310739644 T^{2} + 20591960554472314992 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} + 20591960554472314992 p^{7} T^{5} + 18085310739644 p^{14} T^{6} + 1507504 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1516840 T + 33953560223548 T^{2} + 24308897939228417368 T^{3} + \)\(49\!\cdots\!42\)\( T^{4} + 24308897939228417368 p^{7} T^{5} + 33953560223548 p^{14} T^{6} + 1516840 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 9540936 T + 50943310648948 T^{2} + 79126620358128821320 T^{3} + \)\(92\!\cdots\!54\)\( T^{4} + 79126620358128821320 p^{7} T^{5} + 50943310648948 p^{14} T^{6} + 9540936 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 4587600 T + 71876902198988 T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!02\)\( T^{4} + \)\(26\!\cdots\!56\)\( p^{7} T^{5} + 71876902198988 p^{14} T^{6} + 4587600 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 162376 T + 117629517059132 T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(62\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!52\)\( p^{7} T^{5} + 117629517059132 p^{14} T^{6} - 162376 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2726760 T + 194690723817052 T^{2} - \)\(63\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} - \)\(63\!\cdots\!40\)\( p^{7} T^{5} + 194690723817052 p^{14} T^{6} - 2726760 p^{21} T^{7} + p^{28} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.75060825903734096300968573320, −7.39730437984379411196794066165, −7.04819762563361964131055672915, −6.96874745851996709222460448477, −6.49862442679794338152802829259, −6.14294161918630615937799561146, −6.10760272817023753315945418344, −5.92190611073857743443737248268, −5.59144478979801110039967963927, −4.86774116987952032282240890359, −4.83582716942884055068904074964, −4.73642138639272425883477322782, −4.18576057422522914050477347873, −3.86422028833383344406699397249, −3.61885037787874930740571916117, −3.58908221216538109403995166119, −3.58773252785093622053396630463, −2.89607801726694685997320743965, −2.63749907432341992040861484720, −2.58624879706366709451621588093, −2.20095749563206351032223108134, −1.80755641176517936723678368218, −1.45226278309704077580706379173, −1.38394878597406718025891697376, −1.11736339007987192973342849214, 0, 0, 0, 0, 1.11736339007987192973342849214, 1.38394878597406718025891697376, 1.45226278309704077580706379173, 1.80755641176517936723678368218, 2.20095749563206351032223108134, 2.58624879706366709451621588093, 2.63749907432341992040861484720, 2.89607801726694685997320743965, 3.58773252785093622053396630463, 3.58908221216538109403995166119, 3.61885037787874930740571916117, 3.86422028833383344406699397249, 4.18576057422522914050477347873, 4.73642138639272425883477322782, 4.83582716942884055068904074964, 4.86774116987952032282240890359, 5.59144478979801110039967963927, 5.92190611073857743443737248268, 6.10760272817023753315945418344, 6.14294161918630615937799561146, 6.49862442679794338152802829259, 6.96874745851996709222460448477, 7.04819762563361964131055672915, 7.39730437984379411196794066165, 7.75060825903734096300968573320

Graph of the $Z$-function along the critical line