Properties

Label 4-168e2-1.1-c5e2-0-1
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $726.005$
Root an. cond. $5.19080$
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  + 18·3-s − 98·7-s + 243·9-s + 100·11-s + 540·13-s + 1.15e3·17-s + 2.94e3·19-s − 1.76e3·21-s + 2.68e3·23-s − 1.73e3·25-s + 2.91e3·27-s + 996·29-s + 2.61e3·31-s + 1.80e3·33-s − 3.49e3·37-s + 9.72e3·39-s + 1.70e4·41-s + 1.36e4·43-s + 1.18e4·47-s + 7.20e3·49-s + 2.07e4·51-s + 3.75e4·53-s + 5.29e4·57-s − 6.32e3·59-s + 3.97e4·61-s − 2.38e4·63-s − 2.73e4·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 9-s + 0.249·11-s + 0.886·13-s + 0.966·17-s + 1.87·19-s − 0.872·21-s + 1.05·23-s − 0.554·25-s + 0.769·27-s + 0.219·29-s + 0.488·31-s + 0.287·33-s − 0.419·37-s + 1.02·39-s + 1.58·41-s + 1.12·43-s + 0.781·47-s + 3/7·49-s + 1.11·51-s + 1.83·53-s + 2.16·57-s − 0.236·59-s + 1.36·61-s − 0.755·63-s − 0.745·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(726.005\)
Root analytic conductor: \(5.19080\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.805229612\)
\(L(\frac12)\) \(\approx\) \(5.805229612\)
\(L(\frac{7}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p^{2} T )^{2} \)
7$C_1$ \( ( 1 + p^{2} T )^{2} \)
good5$C_2^2$ \( 1 + 1734 T^{2} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 100 T + 320086 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 540 T + 526462 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1152 T + 3130846 T^{2} - 1152 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 2944 T + 5655798 T^{2} - 2944 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2684 T + 3830734 T^{2} - 2684 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 996 T + 37205902 T^{2} - 996 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 2616 T + 41609662 T^{2} - 2616 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 3492 T + 72298414 T^{2} + 3492 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 17016 T + 272987742 T^{2} - 17016 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 13640 T + 279761990 T^{2} - 13640 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 11832 T + 202334814 T^{2} - 11832 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 37548 T + 17523254 p T^{2} - 37548 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 6320 T + 780841414 T^{2} + 6320 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 39764 T + 1977385070 T^{2} - 39764 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 27376 T - 46144618 T^{2} + 27376 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 35460 T + 3145043502 T^{2} - 35460 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 64188 T + 4387114438 T^{2} - 64188 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 110088 T + 8439555358 T^{2} + 110088 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 5896 T + 7776870614 T^{2} + 5896 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 167160 T + 16792608382 T^{2} - 167160 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 11876 T + 16694014454 T^{2} + 11876 p^{5} T^{3} + p^{10} T^{4} \)
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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

−12.00966253842578642818859549387, −11.95265075213618678779511576109, −11.01380061730178881994259595637, −10.62852697835868727213198955101, −9.805412413462515376952272122002, −9.711733347828222598616793761568, −8.970940462729748614648497991231, −8.822912633980712027772947173624, −7.86958745091277688507382198088, −7.64454775023428536718660796607, −7.00984979063330933664896188521, −6.43318494405745576205044510504, −5.61756169943708083214830773851, −5.21718185677927462142237574311, −3.90876434420072511251789907978, −3.85986523544438859760547088380, −2.91509801000489828593235243029, −2.56819974292697023958284381970, −1.23388709406404342786805464838, −0.871748704563070905586264274815, 0.871748704563070905586264274815, 1.23388709406404342786805464838, 2.56819974292697023958284381970, 2.91509801000489828593235243029, 3.85986523544438859760547088380, 3.90876434420072511251789907978, 5.21718185677927462142237574311, 5.61756169943708083214830773851, 6.43318494405745576205044510504, 7.00984979063330933664896188521, 7.64454775023428536718660796607, 7.86958745091277688507382198088, 8.822912633980712027772947173624, 8.970940462729748614648497991231, 9.711733347828222598616793761568, 9.805412413462515376952272122002, 10.62852697835868727213198955101, 11.01380061730178881994259595637, 11.95265075213618678779511576109, 12.00966253842578642818859549387

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