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Properties

Label 8-2e20-1.1-c19e4-0-1
Degree $8$
Conductor $1048576$
Sign $1$
Analytic cond. $2.87442\times 10^{7}$
Root an. cond. $8.55694$
Motivic weight $19$
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  + 4.92e6·5-s − 2.36e9·9-s − 2.09e10·13-s − 5.61e11·17-s − 1.99e13·25-s + 2.56e13·29-s − 1.31e15·37-s − 3.36e13·41-s − 1.16e16·45-s − 2.18e16·49-s + 9.23e14·53-s − 1.41e17·61-s − 1.03e17·65-s − 9.97e17·73-s + 1.51e18·81-s − 2.76e18·85-s − 1.68e19·89-s − 3.09e19·97-s − 2.79e19·101-s − 6.03e19·109-s − 7.87e19·113-s + 4.96e19·117-s − 1.28e20·121-s − 1.28e20·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.12·5-s − 2.03·9-s − 0.548·13-s − 1.14·17-s − 1.04·25-s + 0.328·29-s − 1.66·37-s − 0.0160·41-s − 2.29·45-s − 1.91·49-s + 0.0384·53-s − 1.54·61-s − 0.618·65-s − 1.98·73-s + 1.12·81-s − 1.29·85-s − 5.08·89-s − 4.13·97-s − 2.54·101-s − 2.66·109-s − 2.46·113-s + 1.11·117-s − 2.10·121-s − 1.54·125-s + 0.370·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1048576\)    =    \(2^{20}\)
Sign: $1$
Analytic conductor: \(2.87442\times 10^{7}\)
Root analytic conductor: \(8.55694\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 1048576,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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 \( 1 \)
good3$C_2^2 \wr C_2$ \( 1 + 87735044 p^{3} T^{2} + 208194733401458 p^{9} T^{4} + 87735044 p^{41} T^{6} + p^{76} T^{8} \)
5$D_{4}$ \( ( 1 - 492476 p T + 763726453262 p^{2} T^{2} - 492476 p^{20} T^{3} + p^{38} T^{4} )^{2} \)
7$C_2^2 \wr C_2$ \( 1 + 3114329068285636 p T^{2} + \)\(97\!\cdots\!18\)\( p^{3} T^{4} + 3114329068285636 p^{39} T^{6} + p^{76} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + \)\(12\!\cdots\!44\)\( T^{2} + \)\(87\!\cdots\!26\)\( p^{2} T^{4} + \)\(12\!\cdots\!44\)\( p^{38} T^{6} + p^{76} T^{8} \)
13$D_{4}$ \( ( 1 + 10477772964 T + \)\(19\!\cdots\!06\)\( p T^{2} + 10477772964 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 16515298556 p T + \)\(79\!\cdots\!38\)\( p^{2} T^{2} + 16515298556 p^{20} T^{3} + p^{38} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + \)\(17\!\cdots\!36\)\( T^{2} + \)\(17\!\cdots\!06\)\( T^{4} + \)\(17\!\cdots\!36\)\( p^{38} T^{6} + p^{76} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + \)\(24\!\cdots\!28\)\( T^{2} + \)\(25\!\cdots\!34\)\( T^{4} + \)\(24\!\cdots\!28\)\( p^{38} T^{6} + p^{76} T^{8} \)
29$D_{4}$ \( ( 1 - 12849798040700 T + \)\(11\!\cdots\!38\)\( T^{2} - 12849798040700 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + \)\(39\!\cdots\!04\)\( T^{2} + \)\(84\!\cdots\!86\)\( T^{4} + \)\(39\!\cdots\!04\)\( p^{38} T^{6} + p^{76} T^{8} \)
37$D_{4}$ \( ( 1 + 659363754335188 T + \)\(12\!\cdots\!82\)\( T^{2} + 659363754335188 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 16842742117580 T + \)\(87\!\cdots\!22\)\( T^{2} + 16842742117580 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 + \)\(30\!\cdots\!28\)\( T^{2} + \)\(30\!\cdots\!94\)\( T^{4} + \)\(30\!\cdots\!28\)\( p^{38} T^{6} + p^{76} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 + \)\(59\!\cdots\!52\)\( T^{2} + \)\(62\!\cdots\!54\)\( T^{4} + \)\(59\!\cdots\!52\)\( p^{38} T^{6} + p^{76} T^{8} \)
53$D_{4}$ \( ( 1 - 461628797159564 T + \)\(88\!\cdots\!58\)\( T^{2} - 461628797159564 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + \)\(67\!\cdots\!36\)\( T^{2} + \)\(41\!\cdots\!66\)\( T^{4} + \)\(67\!\cdots\!36\)\( p^{38} T^{6} + p^{76} T^{8} \)
61$D_{4}$ \( ( 1 + 70634164285459460 T + \)\(17\!\cdots\!82\)\( T^{2} + 70634164285459460 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + \)\(29\!\cdots\!32\)\( T^{2} + \)\(51\!\cdots\!74\)\( T^{4} + \)\(29\!\cdots\!32\)\( p^{38} T^{6} + p^{76} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 + \)\(38\!\cdots\!04\)\( T^{2} + \)\(80\!\cdots\!26\)\( T^{4} + \)\(38\!\cdots\!04\)\( p^{38} T^{6} + p^{76} T^{8} \)
73$D_{4}$ \( ( 1 + 498928095156953196 T + \)\(34\!\cdots\!78\)\( T^{2} + 498928095156953196 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + \)\(20\!\cdots\!96\)\( T^{2} + \)\(21\!\cdots\!26\)\( T^{4} + \)\(20\!\cdots\!96\)\( p^{38} T^{6} + p^{76} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 + \)\(58\!\cdots\!88\)\( T^{2} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(58\!\cdots\!88\)\( p^{38} T^{6} + p^{76} T^{8} \)
89$D_{4}$ \( ( 1 + 8400995340531565580 T + \)\(36\!\cdots\!18\)\( T^{2} + 8400995340531565580 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 15461926424678600028 T + \)\(16\!\cdots\!62\)\( T^{2} + 15461926424678600028 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
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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

−9.310426433203081428160256322141, −8.868764096123755319100396879867, −8.361942253350585871741865313564, −8.294594668818333281522208987207, −8.115098764347317327128283840553, −7.63280826608642401393506306162, −6.82998739612753392023905193729, −6.82648072266379930280851193394, −6.79536199869017824616752544095, −5.99423491216629115124202710227, −5.83222946429321810947313914370, −5.59139511227030972522739745689, −5.29485446514471291254669111505, −5.09599786646884192619486263866, −4.38801122600226606834201941781, −4.07759932695787252824347704829, −4.04946875202719684765355312526, −3.10356448808316274956111879048, −2.99238605920124674033086250964, −2.69543947527406626610121008575, −2.58190181630710700272865188455, −1.89026245240412246883462114803, −1.65875372588820198466889055487, −1.47702575524254578765710734117, −1.05151135356904823157208636827, 0, 0, 0, 0, 1.05151135356904823157208636827, 1.47702575524254578765710734117, 1.65875372588820198466889055487, 1.89026245240412246883462114803, 2.58190181630710700272865188455, 2.69543947527406626610121008575, 2.99238605920124674033086250964, 3.10356448808316274956111879048, 4.04946875202719684765355312526, 4.07759932695787252824347704829, 4.38801122600226606834201941781, 5.09599786646884192619486263866, 5.29485446514471291254669111505, 5.59139511227030972522739745689, 5.83222946429321810947313914370, 5.99423491216629115124202710227, 6.79536199869017824616752544095, 6.82648072266379930280851193394, 6.82998739612753392023905193729, 7.63280826608642401393506306162, 8.115098764347317327128283840553, 8.294594668818333281522208987207, 8.361942253350585871741865313564, 8.868764096123755319100396879867, 9.310426433203081428160256322141

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