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 + 3·9-s + 4·11-s + 4·13-s − 8·17-s − 4·23-s + 2·25-s − 4·27-s − 4·29-s − 8·31-s − 8·33-s + 4·37-s − 8·39-s − 16·41-s − 16·43-s + 8·47-s + 16·51-s + 4·53-s + 16·59-s − 4·61-s + 8·67-s + 8·69-s − 12·71-s − 12·73-s − 4·75-s + 8·79-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.20·11-s + 1.10·13-s − 1.94·17-s − 0.834·23-s + 2/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 1.28·39-s − 2.49·41-s − 2.43·43-s + 1.16·47-s + 2.24·51-s + 0.549·53-s + 2.08·59-s − 0.512·61-s + 0.977·67-s + 0.963·69-s − 1.42·71-s − 1.40·73-s − 0.461·75-s + 0.900·79-s + 5/9·81-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.11457556150614888002847459674, −7.09536912984812111731608418318, −6.87223295395087364552356383330, −6.54452745151342578597740221740, −6.11496701437998752436670479739, −5.98047010335366169344027546632, −5.42394772179762228906445326732, −5.23708612647897533857359546153, −4.79273265910832254025447315884, −4.39302846430666336361843953328, −3.92007417955532563623689185237, −3.84969312264330882203501203089, −3.45821801223985192203586797087, −2.87282502434015661290993280669, −2.02647888119039871467488963636, −2.01741251832874320211315377965, −1.40951223292203075536820948492, −1.01750916340233385417297320711, 0, 0, 1.01750916340233385417297320711, 1.40951223292203075536820948492, 2.01741251832874320211315377965, 2.02647888119039871467488963636, 2.87282502434015661290993280669, 3.45821801223985192203586797087, 3.84969312264330882203501203089, 3.92007417955532563623689185237, 4.39302846430666336361843953328, 4.79273265910832254025447315884, 5.23708612647897533857359546153, 5.42394772179762228906445326732, 5.98047010335366169344027546632, 6.11496701437998752436670479739, 6.54452745151342578597740221740, 6.87223295395087364552356383330, 7.09536912984812111731608418318, 7.11457556150614888002847459674

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