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Properties

Label 4-3e4-1.1-c23e2-0-1
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $910.130$
Root an. cond. $5.49257$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.24e3·2-s − 1.08e7·4-s + 4.68e7·5-s − 2.11e8·7-s − 1.84e10·8-s + 5.81e10·10-s − 1.46e12·11-s + 1.04e13·13-s − 2.63e11·14-s + 5.40e13·16-s + 2.10e14·17-s − 9.07e14·19-s − 5.07e14·20-s − 1.82e15·22-s − 1.01e16·23-s − 3.42e15·25-s + 1.30e16·26-s + 2.29e15·28-s − 1.84e16·29-s − 2.72e17·31-s + 9.51e16·32-s + 2.61e17·34-s − 9.92e15·35-s − 4.78e14·37-s − 1.12e18·38-s − 8.63e17·40-s − 5.55e18·41-s + ⋯
L(s)  = 1  + 0.428·2-s − 1.29·4-s + 0.428·5-s − 0.0405·7-s − 0.758·8-s + 0.183·10-s − 1.55·11-s + 1.62·13-s − 0.0173·14-s + 0.768·16-s + 1.49·17-s − 1.78·19-s − 0.554·20-s − 0.665·22-s − 2.21·23-s − 0.286·25-s + 0.696·26-s + 0.0523·28-s − 0.281·29-s − 1.92·31-s + 0.467·32-s + 0.639·34-s − 0.0173·35-s − 0.000442·37-s − 0.766·38-s − 0.325·40-s − 1.57·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(910.130\)
Root analytic conductor: \(5.49257\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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)$
bad3 \( 1 \)
good2$D_{4}$ \( 1 - 621 p T + 96791 p^{7} T^{2} - 621 p^{24} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 - 9361764 p T + 8977934193886 p^{4} T^{2} - 9361764 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 211963904 T - 175324586455056306 p^{2} T^{2} + 211963904 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 133542942408 p T + \)\(18\!\cdots\!54\)\( p^{2} T^{2} + 133542942408 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 10491654264748 T + \)\(79\!\cdots\!82\)\( p T^{2} - 10491654264748 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 - 12405177148284 p T + \)\(17\!\cdots\!98\)\( p^{2} T^{2} - 12405177148284 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 907382448537944 T + \)\(32\!\cdots\!42\)\( p T^{2} + 907382448537944 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 10116923323892112 T + \)\(67\!\cdots\!94\)\( T^{2} + 10116923323892112 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 + 637198398109548 p T + \)\(81\!\cdots\!38\)\( T^{2} + 637198398109548 p^{24} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 272793622592745488 T + \)\(55\!\cdots\!82\)\( T^{2} + 272793622592745488 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 478995036787364 T + \)\(91\!\cdots\!26\)\( T^{2} + 478995036787364 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 + 5555714961308771412 T + \)\(17\!\cdots\!22\)\( T^{2} + 5555714961308771412 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 1198322147609320040 T - \)\(25\!\cdots\!10\)\( T^{2} + 1198322147609320040 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 26308565672855777280 T + \)\(74\!\cdots\!90\)\( T^{2} + 26308565672855777280 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - 41127501899415224628 T + \)\(64\!\cdots\!74\)\( T^{2} - 41127501899415224628 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(30\!\cdots\!96\)\( T + \)\(11\!\cdots\!58\)\( T^{2} - \)\(30\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - \)\(59\!\cdots\!20\)\( T + \)\(26\!\cdots\!38\)\( T^{2} - \)\(59\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(16\!\cdots\!88\)\( T + \)\(24\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!88\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(38\!\cdots\!76\)\( T + \)\(28\!\cdots\!66\)\( T^{2} - \)\(38\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(28\!\cdots\!28\)\( T + \)\(13\!\cdots\!54\)\( T^{2} + \)\(28\!\cdots\!28\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - 21069388575313284880 T + \)\(88\!\cdots\!78\)\( T^{2} - 21069388575313284880 p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(32\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(43\!\cdots\!64\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(43\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(10\!\cdots\!92\)\( T + \)\(10\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!92\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−14.92902322066841086829643137371, −14.45891548750022560692765292278, −13.49439744827782343828957043602, −13.27293756579095559829966214706, −12.68061762357793411391092254360, −11.67531161131143678855108464763, −10.40523970032618676532142563323, −10.21604421094181240703298504163, −9.113608249295255972936446812426, −8.330296178940767330501082868492, −7.888735371192775187747329419950, −6.37296826846352112413705352049, −5.58428686868129741849929291779, −5.16094898286293167284693698437, −3.85694554063887107580338314261, −3.69710626990218918665613382661, −2.28679376621512791001265773573, −1.43785345789369896871991193766, 0, 0, 1.43785345789369896871991193766, 2.28679376621512791001265773573, 3.69710626990218918665613382661, 3.85694554063887107580338314261, 5.16094898286293167284693698437, 5.58428686868129741849929291779, 6.37296826846352112413705352049, 7.888735371192775187747329419950, 8.330296178940767330501082868492, 9.113608249295255972936446812426, 10.21604421094181240703298504163, 10.40523970032618676532142563323, 11.67531161131143678855108464763, 12.68061762357793411391092254360, 13.27293756579095559829966214706, 13.49439744827782343828957043602, 14.45891548750022560692765292278, 14.92902322066841086829643137371

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