Properties

Label 8-384e4-1.1-c1e4-0-8
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $88.3961$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·9-s + 12·11-s + 8·23-s + 8·25-s − 6·27-s − 24·33-s + 16·37-s + 32·47-s + 16·49-s + 4·59-s − 16·61-s − 16·69-s + 8·71-s − 8·73-s − 16·75-s + 11·81-s + 20·83-s − 16·97-s + 24·99-s − 12·107-s + 16·109-s − 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.15·3-s + 2/3·9-s + 3.61·11-s + 1.66·23-s + 8/5·25-s − 1.15·27-s − 4.17·33-s + 2.63·37-s + 4.66·47-s + 16/7·49-s + 0.520·59-s − 2.04·61-s − 1.92·69-s + 0.949·71-s − 0.936·73-s − 1.84·75-s + 11/9·81-s + 2.19·83-s − 1.62·97-s + 2.41·99-s − 1.16·107-s + 1.53·109-s − 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(88.3961\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.488575444\)
\(L(\frac12)\) \(\approx\) \(2.488575444\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$D_4\times C_2$ \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
11$C_4$ \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
53$D_4\times C_2$ \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$D_4\times C_2$ \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.375311004136550745503923058818, −7.81355643235460799238542660951, −7.44684296800419373091218472860, −7.38768480962131930711332998032, −7.32744672378307609304812102459, −6.93886039116066102818692117218, −6.49958337079848265010822460002, −6.46217092559972383772333794662, −6.31772620663705454161691159052, −5.95528125231371632951653869061, −5.79526940032163107620983650896, −5.36493042952581033648915175603, −5.24017022313894892514145531576, −4.78007082550275247062899098846, −4.39857266292059540737170685877, −4.25027968734526827701723628527, −4.06233545444411783821465339472, −3.61708637713807937652164434514, −3.59307890147807467427570262616, −2.86233646484382091941252710760, −2.41150850783916056386223581536, −2.34162863280202814876175064112, −1.21607159322177550095062969573, −1.11252385058558417766764830712, −0.990795964148147302576711378088, 0.990795964148147302576711378088, 1.11252385058558417766764830712, 1.21607159322177550095062969573, 2.34162863280202814876175064112, 2.41150850783916056386223581536, 2.86233646484382091941252710760, 3.59307890147807467427570262616, 3.61708637713807937652164434514, 4.06233545444411783821465339472, 4.25027968734526827701723628527, 4.39857266292059540737170685877, 4.78007082550275247062899098846, 5.24017022313894892514145531576, 5.36493042952581033648915175603, 5.79526940032163107620983650896, 5.95528125231371632951653869061, 6.31772620663705454161691159052, 6.46217092559972383772333794662, 6.49958337079848265010822460002, 6.93886039116066102818692117218, 7.32744672378307609304812102459, 7.38768480962131930711332998032, 7.44684296800419373091218472860, 7.81355643235460799238542660951, 8.375311004136550745503923058818

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