Properties

Label 4-528e2-1.1-c3e2-0-0
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $970.509$
Root an. cond. $5.58148$
Motivic weight $3$
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  − 6·3-s + 10·5-s + 2·7-s + 27·9-s + 22·11-s + 14·13-s − 60·15-s − 80·17-s + 60·19-s − 12·21-s + 202·23-s − 78·25-s − 108·27-s − 336·29-s − 128·31-s − 132·33-s + 20·35-s + 188·37-s − 84·39-s − 132·41-s − 480·43-s + 270·45-s + 390·47-s + 190·49-s + 480·51-s + 610·53-s + 220·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.107·7-s + 9-s + 0.603·11-s + 0.298·13-s − 1.03·15-s − 1.14·17-s + 0.724·19-s − 0.124·21-s + 1.83·23-s − 0.623·25-s − 0.769·27-s − 2.15·29-s − 0.741·31-s − 0.696·33-s + 0.0965·35-s + 0.835·37-s − 0.344·39-s − 0.502·41-s − 1.70·43-s + 0.894·45-s + 1.21·47-s + 0.553·49-s + 1.31·51-s + 1.58·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(970.509\)
Root analytic conductor: \(5.58148\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.922752775\)
\(L(\frac12)\) \(\approx\) \(1.922752775\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 - 2 p T + 178 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 2 T - 186 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 14 T - 310 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 80 T + 11038 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 60 T + 4918 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 202 T + 32110 T^{2} - 202 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 336 T + 73510 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 128 T + 38846 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 188 T + 10814 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 132 T + 86326 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 480 T + 210406 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 390 T + 244798 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 610 T + 271954 T^{2} - 610 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 372 T + 413926 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 1050 T + 724834 T^{2} - 1050 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 408 T + 618310 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 942 T + 915838 T^{2} + 942 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 740 T + 517622 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1646 T + 1592694 T^{2} - 1646 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 352 T + 912262 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 2036 T + 2421430 T^{2} - 2036 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 336 T + 1378270 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.79923791518562197478608995479, −10.32106181671224006891354244305, −9.696751748412414841279409650582, −9.471265044386246122054148490541, −8.841220902902738480266247241829, −8.838372121242310256999615945403, −7.71782216192310043139630785067, −7.47236083895545825440624569546, −6.78181458618387055041840606151, −6.63911664085380043787909307063, −5.94494775032821322954553884594, −5.59122653580966977473208212136, −5.14472422455963385126594027626, −4.74312128190812460751258512295, −3.77302129245868504052670263625, −3.69160937380298655653912194794, −2.48015777178934071585532535367, −1.91669772964964999557095135150, −1.24659347781936715316694198197, −0.49290973390400650477587686718, 0.49290973390400650477587686718, 1.24659347781936715316694198197, 1.91669772964964999557095135150, 2.48015777178934071585532535367, 3.69160937380298655653912194794, 3.77302129245868504052670263625, 4.74312128190812460751258512295, 5.14472422455963385126594027626, 5.59122653580966977473208212136, 5.94494775032821322954553884594, 6.63911664085380043787909307063, 6.78181458618387055041840606151, 7.47236083895545825440624569546, 7.71782216192310043139630785067, 8.838372121242310256999615945403, 8.841220902902738480266247241829, 9.471265044386246122054148490541, 9.696751748412414841279409650582, 10.32106181671224006891354244305, 10.79923791518562197478608995479

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