Properties

Label 4-3e6-1.1-c7e2-0-1
Degree $4$
Conductor $729$
Sign $1$
Analytic cond. $71.1390$
Root an. cond. $2.90420$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s − 49·4-s + 180·5-s + 700·7-s − 459·8-s + 1.62e3·10-s + 1.08e4·11-s − 5.48e3·13-s + 6.30e3·14-s − 3.77e3·16-s + 1.64e4·17-s + 1.60e4·19-s − 8.82e3·20-s + 9.80e4·22-s + 2.43e4·23-s − 1.31e5·25-s − 4.93e4·26-s − 3.43e4·28-s − 1.43e5·29-s − 3.87e4·31-s − 1.78e5·32-s + 1.47e5·34-s + 1.26e5·35-s + 4.55e5·37-s + 1.44e5·38-s − 8.26e4·40-s + 7.31e5·41-s + ⋯
L(s)  = 1  + 0.795·2-s − 0.382·4-s + 0.643·5-s + 0.771·7-s − 0.316·8-s + 0.512·10-s + 2.46·11-s − 0.691·13-s + 0.613·14-s − 0.230·16-s + 0.810·17-s + 0.535·19-s − 0.246·20-s + 1.96·22-s + 0.417·23-s − 1.68·25-s − 0.550·26-s − 0.295·28-s − 1.09·29-s − 0.233·31-s − 0.961·32-s + 0.644·34-s + 0.496·35-s + 1.47·37-s + 0.426·38-s − 0.204·40-s + 1.65·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(71.1390\)
Root analytic conductor: \(2.90420\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 729,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(4.197670201\)
\(L(\frac12)\) \(\approx\) \(4.197670201\)
\(L(\frac{9}{2})\) 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$D_{4}$ \( 1 - 9 T + 65 p T^{2} - 9 p^{7} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 - 36 p T + 32753 p T^{2} - 36 p^{8} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 - 100 p T + 584961 T^{2} - 100 p^{8} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 90 p^{2} T + 68388367 T^{2} - 90 p^{9} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 5480 T + 57188634 T^{2} + 5480 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 16416 T + 79007650 T^{2} - 16416 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 16024 T + 1645526562 T^{2} - 16024 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 24372 T + 2905606450 T^{2} - 24372 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 143280 T + 39622155718 T^{2} + 143280 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 + 38708 T + 26496803553 T^{2} + 38708 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 455620 T + 173526750366 T^{2} - 455620 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 731880 T + 472932732862 T^{2} - 731880 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 1088840 T + 751093452114 T^{2} + 1088840 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 1561500 T + 1424242194466 T^{2} - 1561500 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 2610468 T + 3933113324245 T^{2} - 2610468 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 1731960 T + 4329510506038 T^{2} - 1731960 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 620192 T + 6303036139818 T^{2} + 620192 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 346600 T + 12137122138146 T^{2} - 346600 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 + 4242240 T + 14648438075182 T^{2} + 4242240 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 3145190 T + 18855097518219 T^{2} + 3145190 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 10110616 T + 57627345888222 T^{2} - 10110616 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 644202 T + 52577539308895 T^{2} - 644202 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 6021000 T + 94843763242558 T^{2} - 6021000 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 4098670 T + 5732921354451 T^{2} - 4098670 p^{7} T^{3} + p^{14} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.16608444330908283915524979356, −14.95067542731888423492008013593, −14.61129536199827440290580813104, −14.36127376828542744405228850374, −13.45216182151439023345047420041, −13.37401805387148146810229708379, −12.07313384729784072575182211767, −11.89724814823951627023821017786, −11.17693146382673135756971322119, −10.06910046714440528236479689783, −9.307514850626282630634831711861, −9.079387441532127855578830158154, −7.80135456897173030435445321789, −7.05535118169219634243906319161, −5.95400044765020892187739885034, −5.35480958655317030384014312271, −4.24993938151823659642408588958, −3.78568715282118347840385501980, −2.05866821596066530584194173257, −1.03339949669651165841796199218, 1.03339949669651165841796199218, 2.05866821596066530584194173257, 3.78568715282118347840385501980, 4.24993938151823659642408588958, 5.35480958655317030384014312271, 5.95400044765020892187739885034, 7.05535118169219634243906319161, 7.80135456897173030435445321789, 9.079387441532127855578830158154, 9.307514850626282630634831711861, 10.06910046714440528236479689783, 11.17693146382673135756971322119, 11.89724814823951627023821017786, 12.07313384729784072575182211767, 13.37401805387148146810229708379, 13.45216182151439023345047420041, 14.36127376828542744405228850374, 14.61129536199827440290580813104, 14.95067542731888423492008013593, 16.16608444330908283915524979356

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