Properties

Label 4-1176e2-1.1-c3e2-0-0
Degree $4$
Conductor $1382976$
Sign $1$
Analytic cond. $4814.45$
Root an. cond. $8.32984$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 9·5-s + 27·9-s + 5·11-s − 19·13-s − 54·15-s − 4·17-s + 117·19-s + 148·23-s − 63·25-s − 108·27-s − 95·29-s − 360·31-s − 30·33-s − 53·37-s + 114·39-s − 170·41-s + 403·43-s + 243·45-s + 368·47-s + 24·51-s + 697·53-s + 45·55-s − 702·57-s − 585·59-s − 1.16e3·61-s − 171·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.804·5-s + 9-s + 0.137·11-s − 0.405·13-s − 0.929·15-s − 0.0570·17-s + 1.41·19-s + 1.34·23-s − 0.503·25-s − 0.769·27-s − 0.608·29-s − 2.08·31-s − 0.158·33-s − 0.235·37-s + 0.468·39-s − 0.647·41-s + 1.42·43-s + 0.804·45-s + 1.14·47-s + 0.0658·51-s + 1.80·53-s + 0.110·55-s − 1.63·57-s − 1.29·59-s − 2.43·61-s − 0.326·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4814.45\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1382976,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.983323221\)
\(L(\frac12)\) \(\approx\) \(1.983323221\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 9 T + 144 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 5 T - 488 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 19 T + 4358 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 7810 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 117 T + 10954 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 148 T + 27790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 95 T + 47878 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 360 T + 91477 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 53 T + 101882 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 170 T + 104162 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 403 T + 107580 T^{2} - 403 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 368 T + 77882 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 697 T + 7850 p T^{2} - 697 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 585 T + 404278 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1160 T + 691382 T^{2} + 1160 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 233 T - 57688 T^{2} - 233 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 616 T + 81466 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 817 T + 443820 T^{2} + 817 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 802 T + 855999 T^{2} + 802 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 283 T + 596860 T^{2} + 283 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1858 T + 2211874 T^{2} + 1858 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1729 T + 2190800 T^{2} - 1729 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.432854040697014490404578537524, −9.418400607660161454095210893326, −8.978795520919080559394544190418, −8.512630975348776760785008683687, −7.63121245161861211513153774091, −7.47773322220385998315867455332, −7.05509481517871601602999263121, −6.83906596495618749021088401769, −5.91790213646041763819795529646, −5.82425884765802671957342047445, −5.47812351039140080937377942163, −5.15361212574179298074357278671, −4.30470578653869946203319952553, −4.29109993477914549749763454074, −3.19802616505864076812963507561, −3.10909520937931433231640881281, −2.00446306170654868220288999231, −1.76469951380628009306305264804, −0.979833689671748119224820366819, −0.42455003023466908121396932041, 0.42455003023466908121396932041, 0.979833689671748119224820366819, 1.76469951380628009306305264804, 2.00446306170654868220288999231, 3.10909520937931433231640881281, 3.19802616505864076812963507561, 4.29109993477914549749763454074, 4.30470578653869946203319952553, 5.15361212574179298074357278671, 5.47812351039140080937377942163, 5.82425884765802671957342047445, 5.91790213646041763819795529646, 6.83906596495618749021088401769, 7.05509481517871601602999263121, 7.47773322220385998315867455332, 7.63121245161861211513153774091, 8.512630975348776760785008683687, 8.978795520919080559394544190418, 9.418400607660161454095210893326, 9.432854040697014490404578537524

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