Properties

Degree $4$
Conductor $88510464$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 3·9-s − 4·11-s + 8·15-s − 12·17-s − 8·19-s + 4·23-s + 4·25-s + 4·27-s − 8·33-s − 8·37-s − 12·41-s + 12·45-s + 8·47-s − 24·51-s + 4·53-s − 16·55-s − 16·57-s + 8·61-s + 8·69-s + 4·71-s − 24·73-s + 8·75-s − 16·79-s + 5·81-s + 8·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s − 1.20·11-s + 2.06·15-s − 2.91·17-s − 1.83·19-s + 0.834·23-s + 4/5·25-s + 0.769·27-s − 1.39·33-s − 1.31·37-s − 1.87·41-s + 1.78·45-s + 1.16·47-s − 3.36·51-s + 0.549·53-s − 2.15·55-s − 2.11·57-s + 1.02·61-s + 0.963·69-s + 0.474·71-s − 2.80·73-s + 0.923·75-s − 1.80·79-s + 5/9·81-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.46498760222159273743634623649, −7.05696988149169534066913431227, −6.77383861972425003498732166397, −6.73819596163184691902359302712, −6.15332806349571614185885363535, −5.92261609293520518516079461205, −5.42270841971141695942205780070, −5.13249332089022206491601609001, −4.76265327364282269072740006234, −4.40438842005853880903787914455, −3.90354672070654207970195668352, −3.80878799446104429527838884298, −2.83912290502608412088540378646, −2.75303967902450405859607680443, −2.34776637748721021389676289526, −2.13598259401682409695721936583, −1.66895673320538261841882971201, −1.37582143557372265510835878909, 0, 0, 1.37582143557372265510835878909, 1.66895673320538261841882971201, 2.13598259401682409695721936583, 2.34776637748721021389676289526, 2.75303967902450405859607680443, 2.83912290502608412088540378646, 3.80878799446104429527838884298, 3.90354672070654207970195668352, 4.40438842005853880903787914455, 4.76265327364282269072740006234, 5.13249332089022206491601609001, 5.42270841971141695942205780070, 5.92261609293520518516079461205, 6.15332806349571614185885363535, 6.73819596163184691902359302712, 6.77383861972425003498732166397, 7.05696988149169534066913431227, 7.46498760222159273743634623649

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