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 − 8·13-s − 8·15-s + 4·17-s + 4·23-s + 4·25-s + 4·27-s + 8·29-s + 8·31-s − 8·33-s + 8·37-s − 16·39-s + 4·41-s − 12·45-s + 8·51-s + 4·53-s + 16·55-s − 8·59-s − 16·61-s + 32·65-s + 8·69-s + 4·71-s + 8·73-s + 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.21·13-s − 2.06·15-s + 0.970·17-s + 0.834·23-s + 4/5·25-s + 0.769·27-s + 1.48·29-s + 1.43·31-s − 1.39·33-s + 1.31·37-s − 2.56·39-s + 0.624·41-s − 1.78·45-s + 1.12·51-s + 0.549·53-s + 2.15·55-s − 1.04·59-s − 2.04·61-s + 3.96·65-s + 0.963·69-s + 0.474·71-s + 0.936·73-s + 0.923·75-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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.64220576847279471667602596047, −7.49101707057877893929357540722, −6.95751750934006569202976601564, −6.85657565528535296252539979893, −6.12446269098568068160013365099, −5.93696313033044284677683541877, −5.21697273857543179777815010258, −4.97086612916181162726844709797, −4.60762156926268485180471166863, −4.50336484316351828986473610751, −3.90518688819548096576189827331, −3.72533619561405333864383288380, −2.97196444295711421216455841529, −2.87847742283912419242878879755, −2.57280140431709732388484774159, −2.36284797443541772802451593487, −1.28865534234266389664378458124, −1.08305726150244079543644938249, 0, 0, 1.08305726150244079543644938249, 1.28865534234266389664378458124, 2.36284797443541772802451593487, 2.57280140431709732388484774159, 2.87847742283912419242878879755, 2.97196444295711421216455841529, 3.72533619561405333864383288380, 3.90518688819548096576189827331, 4.50336484316351828986473610751, 4.60762156926268485180471166863, 4.97086612916181162726844709797, 5.21697273857543179777815010258, 5.93696313033044284677683541877, 6.12446269098568068160013365099, 6.85657565528535296252539979893, 6.95751750934006569202976601564, 7.49101707057877893929357540722, 7.64220576847279471667602596047

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