Properties

Label 4-147e2-1.1-c5e2-0-15
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $555.847$
Root an. cond. $4.85555$
Motivic weight $5$
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  − 3·2-s + 18·3-s + 5·4-s + 33·5-s − 54·6-s − 99·8-s + 243·9-s − 99·10-s − 1.13e3·11-s + 90·12-s − 925·13-s + 594·15-s − 459·16-s + 324·17-s − 729·18-s − 2.31e3·19-s + 165·20-s + 3.41e3·22-s + 1.59e3·23-s − 1.78e3·24-s − 2.38e3·25-s + 2.77e3·26-s + 2.91e3·27-s − 2.21e3·29-s − 1.78e3·30-s − 4.29e3·31-s + 4.36e3·32-s + ⋯
L(s)  = 1  − 0.530·2-s + 1.15·3-s + 5/32·4-s + 0.590·5-s − 0.612·6-s − 0.546·8-s + 9-s − 0.313·10-s − 2.83·11-s + 0.180·12-s − 1.51·13-s + 0.681·15-s − 0.448·16-s + 0.271·17-s − 0.530·18-s − 1.46·19-s + 0.0922·20-s + 1.50·22-s + 0.629·23-s − 0.631·24-s − 0.762·25-s + 0.805·26-s + 0.769·27-s − 0.489·29-s − 0.361·30-s − 0.802·31-s + 0.753·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(555.847\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21609,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 + 3 T + p^{2} T^{2} + 3 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 33 T + 3472 T^{2} - 33 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 1137 T + 645232 T^{2} + 1137 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 925 T + 951450 T^{2} + 925 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 324 T + 1502434 T^{2} - 324 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 2311 T + 6241998 T^{2} + 2311 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 1596 T + 7604206 T^{2} - 1596 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 2217 T + 42125014 T^{2} + 2217 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 4294 T + 25151367 T^{2} + 4294 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 19109 T + 184470078 T^{2} + 19109 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 12858 T + 228491122 T^{2} - 12858 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 23160 T + 570076618 T^{2} - 23160 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 31653 T + 896010526 T^{2} + 31653 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 41097 T + 896994610 T^{2} + 41097 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 42052 T + 1728407262 T^{2} + 42052 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 30763 T + 1698033324 T^{2} - 30763 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 102096 T + 6091648810 T^{2} - 102096 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 28577 T + 2636499108 T^{2} - 28577 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 18464 T + 3332046261 T^{2} + 18464 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 61179 T + 8589312784 T^{2} + 61179 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 29322 T + 11382011794 T^{2} + 29322 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 9791 T + 17134261944 T^{2} - 9791 p^{5} T^{3} + p^{10} T^{4} \)
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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

−12.18531646024113554861615028485, −10.98333057672356782561897038682, −10.75025679854187434642862371755, −10.31637998498592291701011526655, −9.597832961713370677363959296157, −9.460409678122790556063826087154, −8.751786800519158757786823859536, −8.252508069274632845332126271264, −7.65023487465555416172064073946, −7.43894621286793819805019524320, −6.67294266882065078484450509044, −5.80905837720830070576998366880, −5.13406605584572312001463416135, −4.71988752942268134871114721622, −3.61859773359340275618543275722, −2.60443771002624025864218642479, −2.51122278055465735574161802883, −1.82645569143843763439410693965, 0, 0, 1.82645569143843763439410693965, 2.51122278055465735574161802883, 2.60443771002624025864218642479, 3.61859773359340275618543275722, 4.71988752942268134871114721622, 5.13406605584572312001463416135, 5.80905837720830070576998366880, 6.67294266882065078484450509044, 7.43894621286793819805019524320, 7.65023487465555416172064073946, 8.252508069274632845332126271264, 8.751786800519158757786823859536, 9.460409678122790556063826087154, 9.597832961713370677363959296157, 10.31637998498592291701011526655, 10.75025679854187434642862371755, 10.98333057672356782561897038682, 12.18531646024113554861615028485

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