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Properties

Label 4-3e4-1.1-c35e2-0-0
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $4877.01$
Root an. cond. $8.35677$
Motivic weight $35$
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  + 6.09e4·2-s − 3.93e9·4-s + 1.33e12·5-s − 1.20e15·7-s + 1.38e15·8-s + 8.12e16·10-s + 1.47e18·11-s + 3.00e19·13-s − 7.31e19·14-s − 9.38e20·16-s − 6.18e21·17-s − 1.60e22·19-s − 5.24e21·20-s + 8.98e22·22-s − 4.93e23·23-s − 4.39e24·25-s + 1.82e24·26-s + 4.72e24·28-s + 7.88e25·29-s − 1.21e26·31-s − 1.47e26·32-s − 3.76e26·34-s − 1.60e27·35-s + 1.66e27·37-s − 9.77e26·38-s + 1.85e27·40-s + 3.29e28·41-s + ⋯
L(s)  = 1  + 0.328·2-s − 0.114·4-s + 0.781·5-s − 1.95·7-s + 0.217·8-s + 0.256·10-s + 0.879·11-s + 0.962·13-s − 0.641·14-s − 0.794·16-s − 1.81·17-s − 0.671·19-s − 0.0894·20-s + 0.289·22-s − 0.730·23-s − 1.51·25-s + 0.316·26-s + 0.223·28-s + 2.01·29-s − 0.966·31-s − 0.675·32-s − 0.595·34-s − 1.52·35-s + 0.599·37-s − 0.220·38-s + 0.170·40-s + 1.97·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+35/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(4877.01\)
Root analytic conductor: \(8.35677\)
Motivic weight: \(35\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81,\ (\ :35/2, 35/2),\ 1)\)

Particular Values

\(L(18)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{37}{2})\) 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$D_{4}$ \( 1 - 3807 p^{4} T + 3732131 p^{11} T^{2} - 3807 p^{39} T^{3} + p^{70} T^{4} \)
5$D_{4}$ \( 1 - 266755899348 p T + \)\(19\!\cdots\!58\)\( p^{5} T^{2} - 266755899348 p^{36} T^{3} + p^{70} T^{4} \)
7$D_{4}$ \( 1 + 24520669039856 p^{2} T + \)\(46\!\cdots\!86\)\( p^{4} T^{2} + 24520669039856 p^{37} T^{3} + p^{70} T^{4} \)
11$D_{4}$ \( 1 - 1474443852221320632 T + \)\(44\!\cdots\!54\)\( p^{2} T^{2} - 1474443852221320632 p^{35} T^{3} + p^{70} T^{4} \)
13$D_{4}$ \( 1 - 2308093358323513756 p T + \)\(25\!\cdots\!98\)\( p^{3} T^{2} - 2308093358323513756 p^{36} T^{3} + p^{70} T^{4} \)
17$D_{4}$ \( 1 + \)\(36\!\cdots\!96\)\( p T + \)\(10\!\cdots\!78\)\( p^{2} T^{2} + \)\(36\!\cdots\!96\)\( p^{36} T^{3} + p^{70} T^{4} \)
19$D_{4}$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(29\!\cdots\!62\)\( p T^{2} + \)\(16\!\cdots\!64\)\( p^{35} T^{3} + p^{70} T^{4} \)
23$D_{4}$ \( 1 + \)\(49\!\cdots\!72\)\( T + \)\(39\!\cdots\!14\)\( T^{2} + \)\(49\!\cdots\!72\)\( p^{35} T^{3} + p^{70} T^{4} \)
29$D_{4}$ \( 1 - \)\(78\!\cdots\!48\)\( T + \)\(15\!\cdots\!22\)\( p T^{2} - \)\(78\!\cdots\!48\)\( p^{35} T^{3} + p^{70} T^{4} \)
31$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( p^{2} T + \)\(35\!\cdots\!82\)\( p^{2} T^{2} + \)\(12\!\cdots\!68\)\( p^{37} T^{3} + p^{70} T^{4} \)
37$D_{4}$ \( 1 - \)\(16\!\cdots\!16\)\( T + \)\(20\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!16\)\( p^{35} T^{3} + p^{70} T^{4} \)
41$D_{4}$ \( 1 - \)\(32\!\cdots\!88\)\( T + \)\(68\!\cdots\!22\)\( T^{2} - \)\(32\!\cdots\!88\)\( p^{35} T^{3} + p^{70} T^{4} \)
43$D_{4}$ \( 1 - \)\(10\!\cdots\!20\)\( T - \)\(10\!\cdots\!10\)\( T^{2} - \)\(10\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
47$D_{4}$ \( 1 - \)\(52\!\cdots\!20\)\( T + \)\(66\!\cdots\!90\)\( T^{2} - \)\(52\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
53$D_{4}$ \( 1 - \)\(31\!\cdots\!68\)\( T - \)\(16\!\cdots\!26\)\( T^{2} - \)\(31\!\cdots\!68\)\( p^{35} T^{3} + p^{70} T^{4} \)
59$D_{4}$ \( 1 + \)\(82\!\cdots\!64\)\( T + \)\(20\!\cdots\!58\)\( T^{2} + \)\(82\!\cdots\!64\)\( p^{35} T^{3} + p^{70} T^{4} \)
61$D_{4}$ \( 1 + \)\(40\!\cdots\!40\)\( T + \)\(98\!\cdots\!18\)\( T^{2} + \)\(40\!\cdots\!40\)\( p^{35} T^{3} + p^{70} T^{4} \)
67$D_{4}$ \( 1 - \)\(96\!\cdots\!12\)\( T + \)\(11\!\cdots\!22\)\( T^{2} - \)\(96\!\cdots\!12\)\( p^{35} T^{3} + p^{70} T^{4} \)
71$D_{4}$ \( 1 + \)\(44\!\cdots\!24\)\( T + \)\(99\!\cdots\!46\)\( T^{2} + \)\(44\!\cdots\!24\)\( p^{35} T^{3} + p^{70} T^{4} \)
73$D_{4}$ \( 1 + \)\(13\!\cdots\!08\)\( T + \)\(29\!\cdots\!54\)\( T^{2} + \)\(13\!\cdots\!08\)\( p^{35} T^{3} + p^{70} T^{4} \)
79$D_{4}$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(29\!\cdots\!98\)\( T^{2} + \)\(10\!\cdots\!80\)\( p^{35} T^{3} + p^{70} T^{4} \)
83$D_{4}$ \( 1 - \)\(55\!\cdots\!44\)\( T + \)\(36\!\cdots\!82\)\( T^{2} - \)\(55\!\cdots\!44\)\( p^{35} T^{3} + p^{70} T^{4} \)
89$D_{4}$ \( 1 + \)\(24\!\cdots\!36\)\( T + \)\(40\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!36\)\( p^{35} T^{3} + p^{70} T^{4} \)
97$D_{4}$ \( 1 - \)\(83\!\cdots\!52\)\( T + \)\(66\!\cdots\!62\)\( T^{2} - \)\(83\!\cdots\!52\)\( p^{35} T^{3} + p^{70} T^{4} \)
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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

−13.16108124798278510785400787447, −12.83677548795587386273050765265, −11.93384488416048699268085096278, −11.07936031705720471132076953212, −10.43725477438367231578664876947, −9.602257078159874084775061445352, −9.192768614725149102293519769624, −8.677659690171006578038677457303, −7.45971983013880416147319514147, −6.50590542045593124839213189720, −6.25013584523460759762792617496, −5.92478646395502166647711940970, −4.39431550426686313934764077179, −4.24429707416636498262231266001, −3.37282659810807706667843283717, −2.56158535517741608476418135072, −1.99654893361030356007925909906, −1.18479196421201905355383354332, 0, 0, 1.18479196421201905355383354332, 1.99654893361030356007925909906, 2.56158535517741608476418135072, 3.37282659810807706667843283717, 4.24429707416636498262231266001, 4.39431550426686313934764077179, 5.92478646395502166647711940970, 6.25013584523460759762792617496, 6.50590542045593124839213189720, 7.45971983013880416147319514147, 8.677659690171006578038677457303, 9.192768614725149102293519769624, 9.602257078159874084775061445352, 10.43725477438367231578664876947, 11.07936031705720471132076953212, 11.93384488416048699268085096278, 12.83677548795587386273050765265, 13.16108124798278510785400787447

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