Properties

Label 4-3e4-1.1-c25e2-0-0
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $1270.18$
Root an. cond. $5.96990$
Motivic weight $25$
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  + 324·2-s − 2.06e7·4-s − 5.70e8·5-s − 2.96e10·7-s − 2.57e9·8-s − 1.84e11·10-s + 1.81e13·11-s − 1.05e14·13-s − 9.61e12·14-s − 6.92e14·16-s + 4.56e14·17-s + 1.38e16·19-s + 1.18e16·20-s + 5.89e15·22-s − 1.39e17·23-s − 3.27e17·25-s − 3.40e16·26-s + 6.14e17·28-s + 2.50e18·29-s + 1.07e18·31-s − 7.60e17·32-s + 1.48e17·34-s + 1.69e19·35-s − 1.97e18·37-s + 4.50e18·38-s + 1.46e18·40-s + 4.19e20·41-s + ⋯
L(s)  = 1  + 0.0559·2-s − 0.616·4-s − 1.04·5-s − 0.810·7-s − 0.0132·8-s − 0.0584·10-s + 1.74·11-s − 1.25·13-s − 0.0453·14-s − 0.615·16-s + 0.190·17-s + 1.44·19-s + 0.645·20-s + 0.0977·22-s − 1.32·23-s − 1.09·25-s − 0.0700·26-s + 0.500·28-s + 1.31·29-s + 0.245·31-s − 0.116·32-s + 0.0106·34-s + 0.847·35-s − 0.0492·37-s + 0.0805·38-s + 0.0138·40-s + 2.90·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(1270.18\)
Root analytic conductor: \(5.96990\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(1.054236623\)
\(L(\frac12)\) \(\approx\) \(1.054236623\)
\(L(\frac{27}{2})\) 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$D_{4}$ \( 1 - 81 p^{2} T + 40631 p^{9} T^{2} - 81 p^{27} T^{3} + p^{50} T^{4} \)
5$D_{4}$ \( 1 + 570861756 T + 26127791204487934 p^{2} T^{2} + 570861756 p^{25} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 4241055104 p T + 696480056120365134 p^{4} T^{2} + 4241055104 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 - 1653435056232 p T + \)\(19\!\cdots\!14\)\( p^{3} T^{2} - 1653435056232 p^{26} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 105143636679236 T + \)\(10\!\cdots\!98\)\( p T^{2} + 105143636679236 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 - 456987364349436 T + \)\(41\!\cdots\!14\)\( p T^{2} - 456987364349436 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 731444131334056 p T + \)\(24\!\cdots\!98\)\( p^{2} T^{2} - 731444131334056 p^{26} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 + 6058360821754512 p T + \)\(15\!\cdots\!74\)\( p^{2} T^{2} + 6058360821754512 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 - 2502033490260709812 T + \)\(80\!\cdots\!98\)\( T^{2} - 2502033490260709812 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 1075472102492455312 T - \)\(12\!\cdots\!98\)\( T^{2} - 1075472102492455312 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 + 1971470949238312628 T + \)\(32\!\cdots\!94\)\( T^{2} + 1971470949238312628 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 - \)\(41\!\cdots\!68\)\( T + \)\(84\!\cdots\!62\)\( T^{2} - \)\(41\!\cdots\!68\)\( p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - 38930395898259827080 T + \)\(43\!\cdots\!70\)\( T^{2} - 38930395898259827080 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 - \)\(73\!\cdots\!40\)\( T + \)\(14\!\cdots\!10\)\( T^{2} - \)\(73\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 - \)\(84\!\cdots\!64\)\( T + \)\(43\!\cdots\!86\)\( T^{2} - \)\(84\!\cdots\!64\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 + \)\(84\!\cdots\!96\)\( T + \)\(38\!\cdots\!78\)\( T^{2} + \)\(84\!\cdots\!96\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(72\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(13\!\cdots\!56\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 + \)\(12\!\cdots\!84\)\( T + \)\(25\!\cdots\!66\)\( T^{2} + \)\(12\!\cdots\!84\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(84\!\cdots\!76\)\( T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(84\!\cdots\!76\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(12\!\cdots\!80\)\( T + \)\(91\!\cdots\!98\)\( T^{2} - \)\(12\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 - \)\(87\!\cdots\!12\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(87\!\cdots\!12\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 + \)\(21\!\cdots\!44\)\( T + \)\(11\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 + \)\(55\!\cdots\!16\)\( T + \)\(10\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!16\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−15.63873336097921185543997058778, −14.77093588214545654446446926446, −13.96813089087798739572828127917, −13.68107491323425869770770025495, −12.23201780983411527755688966335, −12.13904401583450031387442993937, −11.49606583997323559537793072120, −10.23486938707575233241450797526, −9.400709210290287216166940925236, −9.180741226354052481359183503806, −7.901149392687559199063134880445, −7.38773330123260946890575673582, −6.47198775740922140209715202616, −5.64783095105107578333774861197, −4.26422460989554818025128684485, −4.22706796403521613357784374587, −3.25193321240808968731774238468, −2.30027280479886149476266860066, −1.04707398580710961070066808697, −0.36830474762161133595401064889, 0.36830474762161133595401064889, 1.04707398580710961070066808697, 2.30027280479886149476266860066, 3.25193321240808968731774238468, 4.22706796403521613357784374587, 4.26422460989554818025128684485, 5.64783095105107578333774861197, 6.47198775740922140209715202616, 7.38773330123260946890575673582, 7.901149392687559199063134880445, 9.180741226354052481359183503806, 9.400709210290287216166940925236, 10.23486938707575233241450797526, 11.49606583997323559537793072120, 12.13904401583450031387442993937, 12.23201780983411527755688966335, 13.68107491323425869770770025495, 13.96813089087798739572828127917, 14.77093588214545654446446926446, 15.63873336097921185543997058778

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