Properties

Label 4-147e2-1.1-c3e2-0-6
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $75.2257$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 6·3-s − 11·4-s + 20·5-s − 12·6-s − 36·8-s + 27·9-s + 40·10-s − 20·11-s + 66·12-s + 104·13-s − 120·15-s + 61·16-s + 116·17-s + 54·18-s + 192·19-s − 220·20-s − 40·22-s + 28·23-s + 216·24-s + 148·25-s + 208·26-s − 108·27-s + 296·29-s − 240·30-s − 104·31-s + 358·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1.37·4-s + 1.78·5-s − 0.816·6-s − 1.59·8-s + 9-s + 1.26·10-s − 0.548·11-s + 1.58·12-s + 2.21·13-s − 2.06·15-s + 0.953·16-s + 1.65·17-s + 0.707·18-s + 2.31·19-s − 2.45·20-s − 0.387·22-s + 0.253·23-s + 1.83·24-s + 1.18·25-s + 1.56·26-s − 0.769·27-s + 1.89·29-s − 1.46·30-s − 0.602·31-s + 1.97·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(75.2257\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.459606273\)
\(L(\frac12)\) \(\approx\) \(2.459606273\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 15 T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 4 p T + 252 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 20 T + 1610 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 8 p T + 5848 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 116 T + 9140 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 192 T + 21966 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 28 T + 22962 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 296 T + 62994 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 104 T + 57286 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 248 T + 112074 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 20 T + 42020 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 96 T + 84950 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 56510 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 616 T + 403470 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 454008 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 144 T + 57558 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 988 T + 852210 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 104 T + 459136 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 944 T + 1097470 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1016 T + 1388838 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 388 T + 1217732 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 488 T + 1167280 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \)
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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

−12.89013581185121024513387114812, −12.60153794048880854982947016504, −11.77658331354819561191606962764, −11.66225681106561843424255459710, −10.59945077944475582313314238480, −10.28714456508783178053882430454, −9.768600631814430345034724938467, −9.532896659771571726199669550105, −8.725739071186004366609981559003, −8.331574874294928749970975040208, −7.46077453457776503172401273963, −6.50192839675392189190964801012, −5.93476750785125016658678600706, −5.69489440829089774115211728652, −4.98876506615176824019409913789, −4.97533458659597421169175463349, −3.45661462245391568881073052802, −3.37102006277055577135782932611, −1.45458489799164087560544367544, −0.877718337794786979014334370412, 0.877718337794786979014334370412, 1.45458489799164087560544367544, 3.37102006277055577135782932611, 3.45661462245391568881073052802, 4.97533458659597421169175463349, 4.98876506615176824019409913789, 5.69489440829089774115211728652, 5.93476750785125016658678600706, 6.50192839675392189190964801012, 7.46077453457776503172401273963, 8.331574874294928749970975040208, 8.725739071186004366609981559003, 9.532896659771571726199669550105, 9.768600631814430345034724938467, 10.28714456508783178053882430454, 10.59945077944475582313314238480, 11.66225681106561843424255459710, 11.77658331354819561191606962764, 12.60153794048880854982947016504, 12.89013581185121024513387114812

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