Properties

Label 4-147e2-1.1-c5e2-0-8
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 18·3-s − 9·4-s + 72·5-s + 54·6-s − 15·8-s + 243·9-s − 216·10-s + 480·11-s + 162·12-s + 1.29e3·13-s − 1.29e3·15-s − 529·16-s + 936·17-s − 729·18-s − 216·19-s − 648·20-s − 1.44e3·22-s − 504·23-s + 270·24-s − 2.36e3·25-s − 3.88e3·26-s − 2.91e3·27-s + 6.37e3·29-s + 3.88e3·30-s + 9.93e3·31-s + 5.79e3·32-s + ⋯
L(s)  = 1  − 0.530·2-s − 1.15·3-s − 0.281·4-s + 1.28·5-s + 0.612·6-s − 0.0828·8-s + 9-s − 0.683·10-s + 1.19·11-s + 0.324·12-s + 2.12·13-s − 1.48·15-s − 0.516·16-s + 0.785·17-s − 0.530·18-s − 0.137·19-s − 0.362·20-s − 0.634·22-s − 0.198·23-s + 0.0956·24-s − 0.755·25-s − 1.12·26-s − 0.769·27-s + 1.40·29-s + 0.788·30-s + 1.85·31-s + 1.00·32-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: \(0\)
Selberg data: \((4,\ 21609,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.297463308\)
\(L(\frac12)\) \(\approx\) \(2.297463308\)
\(L(\frac{7}{2})\) 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$D_{4}$ \( 1 + 3 T + 9 p T^{2} + 3 p^{5} T^{3} + p^{10} T^{4} \)
5$C_2$ \( ( 1 - 36 T + p^{5} T^{2} )^{2} \)
11$D_{4}$ \( 1 - 480 T + 376614 T^{2} - 480 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 1296 T + 912362 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 936 T + 807586 T^{2} - 936 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 216 T + 961814 T^{2} + 216 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 504 T + 12784878 T^{2} + 504 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 6372 T + 31409694 T^{2} - 6372 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 9936 T + 65931134 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 11124 T + 115818446 T^{2} - 11124 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 20952 T + 339207826 T^{2} - 20952 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 6264 T + 25906310 T^{2} + 6264 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 7920 T - 101923298 T^{2} - 7920 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 2220 T + 769983534 T^{2} - 2220 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 504 p T + 1614887590 T^{2} - 504 p^{6} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 17280 T + 707051402 T^{2} + 17280 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 20680 T + 11658642 p T^{2} + 20680 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 92280 T + 5423120334 T^{2} + 92280 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 56592 T + 4777720274 T^{2} - 56592 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 56096 T + 4914766302 T^{2} + 56096 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 71352 T + 4792627990 T^{2} + 71352 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 123192 T + 14311603186 T^{2} - 123192 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 35856 T - 1009376254 T^{2} - 35856 p^{5} T^{3} + p^{10} T^{4} \)
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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

−12.19289985383651485720853556257, −11.67161402015328850697049868118, −11.54671236992937161041205407819, −10.83297079356045490295806825367, −10.11836501561123996283428693267, −10.04104152857849535960409107043, −9.323694854004559267706260818217, −8.953705187776048323470145902575, −8.350334454298132775655542657808, −7.73888855441177545298502215249, −6.72840305859930952293190095179, −6.17308660688735500903120277548, −6.10798913122270606452282899625, −5.58783053482362562003933596407, −4.39315914171940491356431612961, −4.21902896428097380529159442164, −3.06159944040201632589000446740, −1.95607142238872516671517008718, −1.04175397356406014468170844194, −0.814533279844596986429365072054, 0.814533279844596986429365072054, 1.04175397356406014468170844194, 1.95607142238872516671517008718, 3.06159944040201632589000446740, 4.21902896428097380529159442164, 4.39315914171940491356431612961, 5.58783053482362562003933596407, 6.10798913122270606452282899625, 6.17308660688735500903120277548, 6.72840305859930952293190095179, 7.73888855441177545298502215249, 8.350334454298132775655542657808, 8.953705187776048323470145902575, 9.323694854004559267706260818217, 10.04104152857849535960409107043, 10.11836501561123996283428693267, 10.83297079356045490295806825367, 11.54671236992937161041205407819, 11.67161402015328850697049868118, 12.19289985383651485720853556257

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