Properties

Label 8-384e4-1.1-c7e4-0-1
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 $0$

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: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.5211772994\)
\(L(\frac12)\) \(\approx\) \(0.5211772994\)
\(L(\frac{9}{2})\) 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

−6.86547010537364016706587089164, −6.62813137866630738971931407415, −6.40608385778696997429258922288, −6.32585497882127143005424700207, −6.12389578883185495841774383008, −5.61175374426131426629663343817, −5.42986958092091300459173422967, −5.35495398909174766470442178143, −5.13243576145879017945941141478, −4.63859509980268730015573761277, −4.46952553616210915080110357620, −4.30076200687962871967469518738, −4.19146190630095356125610430573, −3.39974047809735801793584079340, −3.31651397316410551014258594219, −3.18289653470736433395013161993, −2.68802076081585585974929507644, −2.16875437782085382275267678880, −1.82260947168284919269125794292, −1.79582784521559262644484310502, −1.51520431439735103897057654735, −1.02358244873741824846200723606, −0.55632450157245539509988612310, −0.50124921344335580364819895563, −0.14202851487306519949519794418, 0.14202851487306519949519794418, 0.50124921344335580364819895563, 0.55632450157245539509988612310, 1.02358244873741824846200723606, 1.51520431439735103897057654735, 1.79582784521559262644484310502, 1.82260947168284919269125794292, 2.16875437782085382275267678880, 2.68802076081585585974929507644, 3.18289653470736433395013161993, 3.31651397316410551014258594219, 3.39974047809735801793584079340, 4.19146190630095356125610430573, 4.30076200687962871967469518738, 4.46952553616210915080110357620, 4.63859509980268730015573761277, 5.13243576145879017945941141478, 5.35495398909174766470442178143, 5.42986958092091300459173422967, 5.61175374426131426629663343817, 6.12389578883185495841774383008, 6.32585497882127143005424700207, 6.40608385778696997429258922288, 6.62813137866630738971931407415, 6.86547010537364016706587089164

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