Properties

Label 8-48e4-1.1-c27e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $2.41540\times 10^{9}$
Root an. cond. $14.8892$
Motivic weight $27$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.37e6·3-s + 4.94e9·5-s + 8.82e10·7-s + 2.54e13·9-s + 1.11e14·11-s − 7.33e12·13-s − 3.15e16·15-s + 3.24e16·17-s + 3.28e16·19-s − 5.62e17·21-s + 3.21e17·23-s + 6.24e18·25-s − 8.10e19·27-s + 1.43e20·29-s + 1.52e19·31-s − 7.09e20·33-s + 4.36e20·35-s + 1.11e21·37-s + 4.68e19·39-s + 1.07e22·41-s − 9.88e21·43-s + 1.25e23·45-s − 5.01e22·47-s − 1.33e22·49-s − 2.07e23·51-s + 1.03e23·53-s + 5.50e23·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.81·5-s + 0.344·7-s + 10/3·9-s + 0.971·11-s − 0.00672·13-s − 4.18·15-s + 0.795·17-s + 0.179·19-s − 0.795·21-s + 0.133·23-s + 0.837·25-s − 3.84·27-s + 2.59·29-s + 0.112·31-s − 2.24·33-s + 0.624·35-s + 0.752·37-s + 0.0155·39-s + 1.80·41-s − 0.877·43-s + 6.04·45-s − 1.34·47-s − 0.202·49-s − 1.83·51-s + 0.546·53-s + 1.76·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+27/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.41540\times 10^{9}\)
Root analytic conductor: \(14.8892\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :27/2, 27/2, 27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(7.041653651\)
\(L(\frac12)\) \(\approx\) \(7.041653651\)
\(L(\frac{29}{2})\) 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$C_1$ \( ( 1 + p^{13} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 4949050424 T + 729990698954339564 p^{2} T^{2} - \)\(38\!\cdots\!04\)\( p^{6} T^{3} + \)\(52\!\cdots\!66\)\( p^{8} T^{4} - \)\(38\!\cdots\!04\)\( p^{33} T^{5} + 729990698954339564 p^{56} T^{6} - 4949050424 p^{81} T^{7} + p^{108} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 88249731552 T + \)\(30\!\cdots\!60\)\( p T^{2} + \)\(53\!\cdots\!52\)\( p^{2} T^{3} + \)\(11\!\cdots\!86\)\( p^{5} T^{4} + \)\(53\!\cdots\!52\)\( p^{29} T^{5} + \)\(30\!\cdots\!60\)\( p^{55} T^{6} - 88249731552 p^{81} T^{7} + p^{108} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 111189633716848 T + \)\(45\!\cdots\!72\)\( p T^{2} - \)\(65\!\cdots\!64\)\( p^{3} T^{3} + \)\(12\!\cdots\!30\)\( p^{5} T^{4} - \)\(65\!\cdots\!64\)\( p^{30} T^{5} + \)\(45\!\cdots\!72\)\( p^{55} T^{6} - 111189633716848 p^{81} T^{7} + p^{108} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 7339657642664 T + \)\(28\!\cdots\!20\)\( p T^{2} + \)\(10\!\cdots\!36\)\( p^{2} T^{3} + \)\(27\!\cdots\!30\)\( p^{3} T^{4} + \)\(10\!\cdots\!36\)\( p^{29} T^{5} + \)\(28\!\cdots\!20\)\( p^{55} T^{6} + 7339657642664 p^{81} T^{7} + p^{108} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 32484737251067464 T + \)\(13\!\cdots\!16\)\( p T^{2} - \)\(40\!\cdots\!00\)\( p^{2} T^{3} + \)\(48\!\cdots\!46\)\( p^{3} T^{4} - \)\(40\!\cdots\!00\)\( p^{29} T^{5} + \)\(13\!\cdots\!16\)\( p^{55} T^{6} - 32484737251067464 p^{81} T^{7} + p^{108} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 32881723696992400 T + \)\(48\!\cdots\!08\)\( p T^{2} - \)\(29\!\cdots\!44\)\( p^{2} T^{3} + \)\(59\!\cdots\!94\)\( p^{3} T^{4} - \)\(29\!\cdots\!44\)\( p^{29} T^{5} + \)\(48\!\cdots\!08\)\( p^{55} T^{6} - 32881723696992400 p^{81} T^{7} + p^{108} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 13978313034209312 p T + \)\(15\!\cdots\!80\)\( p^{2} T^{2} - \)\(68\!\cdots\!92\)\( p^{3} T^{3} + \)\(11\!\cdots\!42\)\( p^{4} T^{4} - \)\(68\!\cdots\!92\)\( p^{30} T^{5} + \)\(15\!\cdots\!80\)\( p^{56} T^{6} - 13978313034209312 p^{82} T^{7} + p^{108} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!08\)\( T + \)\(14\!\cdots\!60\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} - \)\(10\!\cdots\!48\)\( p^{27} T^{5} + \)\(14\!\cdots\!60\)\( p^{54} T^{6} - \)\(14\!\cdots\!08\)\( p^{81} T^{7} + p^{108} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 15263052153492355648 T + \)\(39\!\cdots\!84\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(77\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!84\)\( p^{27} T^{5} + \)\(39\!\cdots\!84\)\( p^{54} T^{6} - 15263052153492355648 p^{81} T^{7} + p^{108} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(11\!\cdots\!16\)\( T + \)\(74\!\cdots\!92\)\( T^{2} - \)\(57\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!94\)\( T^{4} - \)\(57\!\cdots\!84\)\( p^{27} T^{5} + \)\(74\!\cdots\!92\)\( p^{54} T^{6} - \)\(11\!\cdots\!16\)\( p^{81} T^{7} + p^{108} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(13\!\cdots\!80\)\( T^{2} - \)\(95\!\cdots\!36\)\( T^{3} + \)\(73\!\cdots\!30\)\( T^{4} - \)\(95\!\cdots\!36\)\( p^{27} T^{5} + \)\(13\!\cdots\!80\)\( p^{54} T^{6} - \)\(10\!\cdots\!80\)\( p^{81} T^{7} + p^{108} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(98\!\cdots\!08\)\( T + \)\(31\!\cdots\!08\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!38\)\( T^{4} + \)\(14\!\cdots\!80\)\( p^{27} T^{5} + \)\(31\!\cdots\!08\)\( p^{54} T^{6} + \)\(98\!\cdots\!08\)\( p^{81} T^{7} + p^{108} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(50\!\cdots\!76\)\( T + \)\(61\!\cdots\!72\)\( T^{2} + \)\(20\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!34\)\( T^{4} + \)\(20\!\cdots\!64\)\( p^{27} T^{5} + \)\(61\!\cdots\!72\)\( p^{54} T^{6} + \)\(50\!\cdots\!76\)\( p^{81} T^{7} + p^{108} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!88\)\( T + \)\(49\!\cdots\!88\)\( T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!74\)\( T^{4} - \)\(25\!\cdots\!68\)\( p^{27} T^{5} + \)\(49\!\cdots\!88\)\( p^{54} T^{6} - \)\(10\!\cdots\!88\)\( p^{81} T^{7} + p^{108} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(19\!\cdots\!24\)\( T + \)\(25\!\cdots\!48\)\( T^{2} - \)\(24\!\cdots\!48\)\( T^{3} + \)\(21\!\cdots\!50\)\( T^{4} - \)\(24\!\cdots\!48\)\( p^{27} T^{5} + \)\(25\!\cdots\!48\)\( p^{54} T^{6} - \)\(19\!\cdots\!24\)\( p^{81} T^{7} + p^{108} T^{8} \)
61$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(40\!\cdots\!20\)\( T^{2} + \)\(40\!\cdots\!24\)\( T^{3} + \)\(90\!\cdots\!50\)\( T^{4} + \)\(40\!\cdots\!24\)\( p^{27} T^{5} + \)\(40\!\cdots\!20\)\( p^{54} T^{6} + \)\(10\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(26\!\cdots\!80\)\( p T + \)\(42\!\cdots\!36\)\( T^{2} - \)\(36\!\cdots\!52\)\( T^{3} + \)\(93\!\cdots\!50\)\( T^{4} - \)\(36\!\cdots\!52\)\( p^{27} T^{5} + \)\(42\!\cdots\!36\)\( p^{54} T^{6} - \)\(26\!\cdots\!80\)\( p^{82} T^{7} + p^{108} T^{8} \)
71$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!36\)\( T + \)\(26\!\cdots\!56\)\( T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(32\!\cdots\!78\)\( T^{4} + \)\(27\!\cdots\!08\)\( p^{27} T^{5} + \)\(26\!\cdots\!56\)\( p^{54} T^{6} + \)\(10\!\cdots\!36\)\( p^{81} T^{7} + p^{108} T^{8} \)
73$C_2 \wr S_4$ \( 1 - \)\(32\!\cdots\!48\)\( T + \)\(70\!\cdots\!52\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!80\)\( p^{27} T^{5} + \)\(70\!\cdots\!52\)\( p^{54} T^{6} - \)\(32\!\cdots\!48\)\( p^{81} T^{7} + p^{108} T^{8} \)
79$C_2 \wr S_4$ \( 1 + \)\(12\!\cdots\!12\)\( T + \)\(10\!\cdots\!40\)\( T^{2} + \)\(63\!\cdots\!32\)\( T^{3} + \)\(31\!\cdots\!14\)\( T^{4} + \)\(63\!\cdots\!32\)\( p^{27} T^{5} + \)\(10\!\cdots\!40\)\( p^{54} T^{6} + \)\(12\!\cdots\!12\)\( p^{81} T^{7} + p^{108} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(79\!\cdots\!64\)\( T^{2} + \)\(50\!\cdots\!36\)\( T^{3} + \)\(79\!\cdots\!50\)\( T^{4} + \)\(50\!\cdots\!36\)\( p^{27} T^{5} + \)\(79\!\cdots\!64\)\( p^{54} T^{6} + \)\(66\!\cdots\!20\)\( p^{81} T^{7} + p^{108} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!08\)\( T + \)\(16\!\cdots\!64\)\( T^{2} + \)\(22\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!72\)\( p^{27} T^{5} + \)\(16\!\cdots\!64\)\( p^{54} T^{6} + \)\(18\!\cdots\!08\)\( p^{81} T^{7} + p^{108} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(41\!\cdots\!92\)\( T^{2} - \)\(54\!\cdots\!76\)\( T^{3} + \)\(75\!\cdots\!78\)\( T^{4} - \)\(54\!\cdots\!76\)\( p^{27} T^{5} + \)\(41\!\cdots\!92\)\( p^{54} T^{6} - \)\(10\!\cdots\!72\)\( p^{81} T^{7} + p^{108} T^{8} \)
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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

−6.87454731170604583470044662478, −6.39005105666963346742910720730, −6.28481499833877547460986629064, −6.18561020687302061466224183345, −6.06978738294251731777489764114, −5.58793050232195868164245456329, −5.32740790409612629353785811329, −5.20170723710639530601610185866, −4.97957190312824053769100598158, −4.55179768621701084531364368105, −4.23935200993842494788153466615, −4.07033695422439326009947309723, −3.97414045194527973308909981167, −3.07960985486359999548074904911, −2.96996115531899658075798460700, −2.92406161847869890721194748666, −2.32508424277773750676866833109, −1.93998130108410027598658281305, −1.68249720338579541644636566031, −1.57626118585058910717583850251, −1.33741138524817812724328805460, −0.76587562504596423637576007514, −0.76507524631649378051301296455, −0.70455644196356570856753195785, −0.25130397840886099399549472347, 0.25130397840886099399549472347, 0.70455644196356570856753195785, 0.76507524631649378051301296455, 0.76587562504596423637576007514, 1.33741138524817812724328805460, 1.57626118585058910717583850251, 1.68249720338579541644636566031, 1.93998130108410027598658281305, 2.32508424277773750676866833109, 2.92406161847869890721194748666, 2.96996115531899658075798460700, 3.07960985486359999548074904911, 3.97414045194527973308909981167, 4.07033695422439326009947309723, 4.23935200993842494788153466615, 4.55179768621701084531364368105, 4.97957190312824053769100598158, 5.20170723710639530601610185866, 5.32740790409612629353785811329, 5.58793050232195868164245456329, 6.06978738294251731777489764114, 6.18561020687302061466224183345, 6.28481499833877547460986629064, 6.39005105666963346742910720730, 6.87454731170604583470044662478

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