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Properties

Label 4-3e4-1.1-c31e2-0-0
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $3001.88$
Root an. cond. $7.40198$
Motivic weight $31$
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  + 3.95e4·2-s − 1.46e9·4-s + 7.93e9·5-s − 1.04e13·7-s − 9.28e13·8-s + 3.13e14·10-s − 1.07e16·11-s − 9.16e16·13-s − 4.14e17·14-s − 4.82e17·16-s + 1.65e19·17-s − 8.24e19·19-s − 1.16e19·20-s − 4.26e20·22-s + 3.51e21·23-s − 6.96e21·25-s − 3.62e21·26-s + 1.53e22·28-s − 2.72e22·29-s + 3.26e22·31-s − 4.46e22·32-s + 6.55e23·34-s − 8.31e22·35-s − 3.58e24·37-s − 3.26e24·38-s − 7.36e23·40-s + 3.79e24·41-s + ⋯
L(s)  = 1  + 0.852·2-s − 0.683·4-s + 0.116·5-s − 0.835·7-s − 0.932·8-s + 0.0991·10-s − 0.778·11-s − 0.496·13-s − 0.712·14-s − 0.104·16-s + 1.40·17-s − 1.24·19-s − 0.0793·20-s − 0.663·22-s + 2.74·23-s − 1.49·25-s − 0.423·26-s + 0.570·28-s − 0.587·29-s + 0.249·31-s − 0.208·32-s + 1.19·34-s − 0.0970·35-s − 1.76·37-s − 1.06·38-s − 0.108·40-s + 0.381·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(3001.88\)
Root analytic conductor: \(7.40198\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{33}{2})\) 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$D_{4}$ \( 1 - 4941 p^{3} T + 2958397 p^{10} T^{2} - 4941 p^{34} T^{3} + p^{62} T^{4} \)
5$D_{4}$ \( 1 - 1586103444 p T + 2247420944432827814 p^{5} T^{2} - 1586103444 p^{32} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 + 1498410605168 p T + \)\(12\!\cdots\!46\)\( p^{4} T^{2} + 1498410605168 p^{32} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 980187318266328 p T + \)\(88\!\cdots\!34\)\( p^{2} T^{2} + 980187318266328 p^{32} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 + 7052829002619716 p T + \)\(26\!\cdots\!78\)\( p^{3} T^{2} + 7052829002619716 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 - 16595285055794710812 T + \)\(17\!\cdots\!06\)\( p T^{2} - 16595285055794710812 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 82480245798578318024 T + \)\(18\!\cdots\!38\)\( p^{2} T^{2} + 82480245798578318024 p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 - \)\(35\!\cdots\!12\)\( T + \)\(26\!\cdots\!58\)\( p T^{2} - \)\(35\!\cdots\!12\)\( p^{31} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 + \)\(94\!\cdots\!28\)\( p T + \)\(43\!\cdots\!38\)\( p^{2} T^{2} + \)\(94\!\cdots\!28\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 - \)\(32\!\cdots\!92\)\( T - \)\(15\!\cdots\!38\)\( T^{2} - \)\(32\!\cdots\!92\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(35\!\cdots\!16\)\( T + \)\(84\!\cdots\!46\)\( T^{2} + \)\(35\!\cdots\!16\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 - \)\(37\!\cdots\!08\)\( T + \)\(20\!\cdots\!82\)\( T^{2} - \)\(37\!\cdots\!08\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(26\!\cdots\!60\)\( T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(26\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(12\!\cdots\!90\)\( T^{2} + \)\(35\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 + \)\(20\!\cdots\!08\)\( T + \)\(33\!\cdots\!34\)\( T^{2} + \)\(20\!\cdots\!08\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 + \)\(83\!\cdots\!44\)\( T + \)\(31\!\cdots\!98\)\( T^{2} + \)\(83\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 - \)\(71\!\cdots\!40\)\( p T + \)\(48\!\cdots\!38\)\( T^{2} - \)\(71\!\cdots\!40\)\( p^{32} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(42\!\cdots\!72\)\( T + \)\(10\!\cdots\!62\)\( T^{2} + \)\(42\!\cdots\!72\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 - \)\(16\!\cdots\!96\)\( T + \)\(29\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!96\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 + \)\(11\!\cdots\!52\)\( T + \)\(12\!\cdots\!34\)\( T^{2} + \)\(11\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(44\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 - \)\(12\!\cdots\!96\)\( T + \)\(97\!\cdots\!82\)\( T^{2} - \)\(12\!\cdots\!96\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 - \)\(46\!\cdots\!04\)\( T + \)\(10\!\cdots\!98\)\( T^{2} - \)\(46\!\cdots\!04\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(96\!\cdots\!88\)\( T + \)\(97\!\cdots\!42\)\( T^{2} - \)\(96\!\cdots\!88\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−13.36442218980056332219046901656, −13.35019837856969012773689029932, −12.57867069339070787119514369793, −12.08648812762420560840747601225, −10.85608672854866864578640426297, −10.34006142604122182043451020793, −9.302581713946890115803615478929, −9.135660939463802572590288755684, −7.951159550887908712966948334149, −7.30796188901086139241117835674, −6.32933390649587539970756819154, −5.61538951461016745835405732066, −4.96186850631289931487325636953, −4.46413284157640019444736334717, −3.33941911316711251450502665424, −3.21928237657074592265994566312, −2.12960607619664671654134735745, −1.15604408468129285222291814463, 0, 0, 1.15604408468129285222291814463, 2.12960607619664671654134735745, 3.21928237657074592265994566312, 3.33941911316711251450502665424, 4.46413284157640019444736334717, 4.96186850631289931487325636953, 5.61538951461016745835405732066, 6.32933390649587539970756819154, 7.30796188901086139241117835674, 7.951159550887908712966948334149, 9.135660939463802572590288755684, 9.302581713946890115803615478929, 10.34006142604122182043451020793, 10.85608672854866864578640426297, 12.08648812762420560840747601225, 12.57867069339070787119514369793, 13.35019837856969012773689029932, 13.36442218980056332219046901656

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