Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 - 1729 T + 2190800 T^{2} - 1729 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.361026282241550955767731889188, −8.255237690608461120930876860555, −7.64573270883740923009935760262, −7.53097614491260272531074617683, −6.77031794688048113931296091401, −6.60510476145190263366415038430, −6.08598502198048380891107114012, −5.89799710238499290538794189969, −5.13505169401371739889768265095, −4.88101163399044358715771657050, −4.31717376280979203520723533958, −3.95124931337291476430299127191, −3.56296518763115254373628572880, −2.90269520919566328380165373338, −2.41956805084769586497092803025, −2.27052620838613659077811634949, −1.52779039632218138644907005679, −1.33907039418938657415600937837, 0, 0, 1.33907039418938657415600937837, 1.52779039632218138644907005679, 2.27052620838613659077811634949, 2.41956805084769586497092803025, 2.90269520919566328380165373338, 3.56296518763115254373628572880, 3.95124931337291476430299127191, 4.31717376280979203520723533958, 4.88101163399044358715771657050, 5.13505169401371739889768265095, 5.89799710238499290538794189969, 6.08598502198048380891107114012, 6.60510476145190263366415038430, 6.77031794688048113931296091401, 7.53097614491260272531074617683, 7.64573270883740923009935760262, 8.255237690608461120930876860555, 8.361026282241550955767731889188

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