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Properties

Label 4-80e2-1.1-c21e2-0-0
Degree $4$
Conductor $6400$
Sign $1$
Analytic cond. $49988.8$
Root an. cond. $14.9526$
Motivic weight $21$
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  − 3.09e4·3-s − 1.95e7·5-s + 4.39e8·7-s − 3.21e9·9-s − 1.05e11·11-s + 3.08e11·13-s + 6.04e11·15-s + 1.83e13·17-s − 2.38e13·19-s − 1.36e13·21-s + 8.25e13·23-s + 2.86e14·25-s − 9.51e13·27-s − 1.67e14·29-s − 5.37e15·31-s + 3.25e15·33-s − 8.59e15·35-s + 8.37e16·37-s − 9.55e15·39-s − 6.32e16·41-s + 1.13e17·43-s + 6.27e16·45-s − 1.32e16·47-s − 9.71e17·49-s − 5.69e17·51-s − 1.44e18·53-s + 2.05e18·55-s + ⋯
L(s)  = 1  − 0.302·3-s − 0.894·5-s + 0.588·7-s − 0.307·9-s − 1.22·11-s + 0.620·13-s + 0.270·15-s + 2.21·17-s − 0.891·19-s − 0.178·21-s + 0.415·23-s + 3/5·25-s − 0.0889·27-s − 0.0737·29-s − 1.17·31-s + 0.370·33-s − 0.526·35-s + 2.86·37-s − 0.187·39-s − 0.735·41-s + 0.803·43-s + 0.274·45-s − 0.0367·47-s − 1.73·49-s − 0.670·51-s − 1.13·53-s + 1.09·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(49988.8\)
Root analytic conductor: \(14.9526\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6400,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(2.156805177\)
\(L(\frac12)\) \(\approx\) \(2.156805177\)
\(L(\frac{23}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p^{10} T )^{2} \)
good3$D_{4}$ \( 1 + 10324 p T + 17175214 p^{5} T^{2} + 10324 p^{22} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 - 439959356 T + 166460827342795614 p T^{2} - 439959356 p^{21} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 + 105191777184 T + \)\(74\!\cdots\!26\)\( p T^{2} + 105191777184 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 - 308456648932 T + \)\(31\!\cdots\!14\)\( p T^{2} - 308456648932 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 - 1081999770612 p T + \)\(72\!\cdots\!42\)\( p^{2} T^{2} - 1081999770612 p^{22} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 + 1253748248200 p T + \)\(28\!\cdots\!58\)\( p^{2} T^{2} + 1253748248200 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 - 82586042978868 T + \)\(77\!\cdots\!02\)\( T^{2} - 82586042978868 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 + 167038888446420 T + \)\(27\!\cdots\!58\)\( T^{2} + 167038888446420 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 + 5373084998145784 T + \)\(46\!\cdots\!26\)\( T^{2} + 5373084998145784 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 83725084127912164 T + \)\(32\!\cdots\!98\)\( T^{2} - 83725084127912164 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 + 63222375005514036 T + \)\(15\!\cdots\!06\)\( T^{2} + 63222375005514036 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 - 113835841911164948 T + \)\(12\!\cdots\!62\)\( T^{2} - 113835841911164948 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 + 13253549001226164 T + \)\(24\!\cdots\!18\)\( T^{2} + 13253549001226164 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 + 1445751023743904748 T + \)\(67\!\cdots\!82\)\( T^{2} + 1445751023743904748 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 817060118931432840 T + \)\(30\!\cdots\!18\)\( T^{2} - 817060118931432840 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 + 4580997169825849436 T + \)\(32\!\cdots\!46\)\( T^{2} + 4580997169825849436 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 - 28808031668773210556 T + \)\(64\!\cdots\!18\)\( T^{2} - 28808031668773210556 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 + 51140957732016114984 T + \)\(21\!\cdots\!06\)\( T^{2} + 51140957732016114984 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 14792322512082321412 T + \)\(38\!\cdots\!82\)\( T^{2} - 14792322512082321412 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 + 25077766877525032720 T + \)\(11\!\cdots\!58\)\( T^{2} + 25077766877525032720 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 - \)\(12\!\cdots\!68\)\( T + \)\(38\!\cdots\!22\)\( T^{2} - \)\(12\!\cdots\!68\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(10\!\cdots\!80\)\( T - \)\(33\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(26\!\cdots\!96\)\( T + \)\(45\!\cdots\!98\)\( T^{2} + \)\(26\!\cdots\!96\)\( p^{21} T^{3} + p^{42} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.05607023216971651467458069116, −10.38051331102989519490097030611, −9.753814254243945551462564225648, −9.285288101267410545192719918171, −8.346183674292668984613501733424, −8.156566112380159963340942345997, −7.67756037489016901414756718171, −7.35023535203193732830069889056, −6.41231007304462841593235643993, −5.92266127343994840601094574329, −5.38231311139979078027807154754, −4.95546869216508336769585692923, −4.30037076847374760581984611025, −3.80195553627326142863721600335, −2.96986903983968927230016128413, −2.94962078199833297198356458408, −1.90510772574495630555294512553, −1.41590804079195700728178142767, −0.71646660031784887901930839687, −0.37337477386618608206128329456, 0.37337477386618608206128329456, 0.71646660031784887901930839687, 1.41590804079195700728178142767, 1.90510772574495630555294512553, 2.94962078199833297198356458408, 2.96986903983968927230016128413, 3.80195553627326142863721600335, 4.30037076847374760581984611025, 4.95546869216508336769585692923, 5.38231311139979078027807154754, 5.92266127343994840601094574329, 6.41231007304462841593235643993, 7.35023535203193732830069889056, 7.67756037489016901414756718171, 8.156566112380159963340942345997, 8.346183674292668984613501733424, 9.285288101267410545192719918171, 9.753814254243945551462564225648, 10.38051331102989519490097030611, 11.05607023216971651467458069116

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