Properties

Label 8-4704e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.896\times 10^{14}$
Sign $1$
Analytic cond. $1.99057\times 10^{6}$
Root an. cond. $6.12875$
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  − 2·9-s − 4·17-s + 20·23-s + 12·25-s + 8·31-s + 4·41-s − 36·71-s − 48·73-s + 8·79-s + 3·81-s − 20·89-s + 32·97-s + 24·103-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2/3·9-s − 0.970·17-s + 4.17·23-s + 12/5·25-s + 1.43·31-s + 0.624·41-s − 4.27·71-s − 5.61·73-s + 0.900·79-s + 1/3·81-s − 2.11·89-s + 3.24·97-s + 2.36·103-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.99057\times 10^{6}\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.278835506\)
\(L(\frac12)\) \(\approx\) \(3.278835506\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 10 T + 68 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 300 T^{2} + 36086 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 10 T + 200 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.95052219106961059175235023150, −5.60189006570404819953216229041, −5.53812911921192988482297313241, −5.23373329469267946722818597361, −4.91138809342765049035436810025, −4.87397685113089965979128691745, −4.61553261189165064108751775504, −4.58851975239867357942218280485, −4.47825969293493488167028834391, −4.13588603613441217181197733588, −4.02805058689944312482706158185, −3.34275055849705788941246843757, −3.29025857509930432691082022509, −3.13925427812280745213278525167, −3.11158346779087527961952163799, −2.84585299933436266808038559654, −2.65658565179589186960588173441, −2.29253823753260603891475996425, −2.28667686361663449145050828473, −1.61548851523429077976484204644, −1.50314050946205367279436635983, −1.17628445191901622172309120096, −0.977826811324644584525613073324, −0.70029963210473911999286094510, −0.25551544004191191850519416008, 0.25551544004191191850519416008, 0.70029963210473911999286094510, 0.977826811324644584525613073324, 1.17628445191901622172309120096, 1.50314050946205367279436635983, 1.61548851523429077976484204644, 2.28667686361663449145050828473, 2.29253823753260603891475996425, 2.65658565179589186960588173441, 2.84585299933436266808038559654, 3.11158346779087527961952163799, 3.13925427812280745213278525167, 3.29025857509930432691082022509, 3.34275055849705788941246843757, 4.02805058689944312482706158185, 4.13588603613441217181197733588, 4.47825969293493488167028834391, 4.58851975239867357942218280485, 4.61553261189165064108751775504, 4.87397685113089965979128691745, 4.91138809342765049035436810025, 5.23373329469267946722818597361, 5.53812911921192988482297313241, 5.60189006570404819953216229041, 5.95052219106961059175235023150

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