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 − 12·5-s + 27·9-s + 4·11-s − 48·13-s + 72·15-s − 84·17-s − 72·19-s + 308·23-s − 44·25-s − 108·27-s − 80·29-s + 384·31-s − 24·33-s + 536·37-s + 288·39-s − 756·41-s − 400·43-s − 324·45-s + 312·47-s + 504·51-s − 52·53-s − 48·55-s + 432·57-s + 864·59-s − 1.41e3·61-s + 576·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.07·5-s + 9-s + 0.109·11-s − 1.02·13-s + 1.23·15-s − 1.19·17-s − 0.869·19-s + 2.79·23-s − 0.351·25-s − 0.769·27-s − 0.512·29-s + 2.22·31-s − 0.126·33-s + 2.38·37-s + 1.18·39-s − 2.87·41-s − 1.41·43-s − 1.07·45-s + 0.968·47-s + 1.38·51-s − 0.134·53-s − 0.117·55-s + 1.00·57-s + 1.90·59-s − 2.97·61-s + 1.09·65-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$D_{4}$ \( 1 + 12 T + 188 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 4 T - 862 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4088 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 84 T + 676 p T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 72 T + 14622 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 308 T + 44522 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 80 T + 18626 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 384 T + 92918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 536 T + 159018 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 756 T + 278276 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 400 T + 142566 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 312 T + 222182 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 52 T + 241982 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 864 T + 596990 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1416 T + 938664 T^{2} + 1416 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 144 T + 550262 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1524 T + 1208266 T^{2} - 1524 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 744 T + 241296 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 976 T + 532734 T^{2} + 976 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 108 T + 1330436 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 744 T + 432480 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.355477389331356983161528638305, −8.139691668642068280721660817580, −7.46731798047526092276988158347, −7.24485555270194383425381139932, −6.89176886707191456339329952219, −6.59174989804686355033151921664, −6.10531294291041106736703152593, −5.82953474884432099329149565301, −4.94115097447557784863637349984, −4.92638340205927121163499983070, −4.43522170484309916812784729217, −4.38825985099628720635506825634, −3.35409584433718378038573501316, −3.34344178470066551648020145337, −2.44926870063739286823686605119, −2.14828663848934822907413031186, −1.22093617224881183511012659946, −0.845022109851300899617793417665, 0, 0, 0.845022109851300899617793417665, 1.22093617224881183511012659946, 2.14828663848934822907413031186, 2.44926870063739286823686605119, 3.34344178470066551648020145337, 3.35409584433718378038573501316, 4.38825985099628720635506825634, 4.43522170484309916812784729217, 4.92638340205927121163499983070, 4.94115097447557784863637349984, 5.82953474884432099329149565301, 6.10531294291041106736703152593, 6.59174989804686355033151921664, 6.89176886707191456339329952219, 7.24485555270194383425381139932, 7.46731798047526092276988158347, 8.139691668642068280721660817580, 8.355477389331356983161528638305

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