Properties

Label 8-168e4-1.1-c1e4-0-0
Degree $8$
Conductor $796594176$
Sign $1$
Analytic cond. $3.23851$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 2·4-s + 6·5-s − 2·7-s + 21·9-s − 6·11-s + 12·12-s − 36·15-s − 12·20-s + 12·21-s + 17·25-s − 54·27-s + 4·28-s − 36·29-s − 30·31-s + 36·33-s − 12·35-s − 42·36-s + 12·44-s + 126·45-s + 7·49-s − 6·53-s − 36·55-s + 18·59-s + 72·60-s − 42·63-s + 8·64-s + ⋯
L(s)  = 1  − 3.46·3-s − 4-s + 2.68·5-s − 0.755·7-s + 7·9-s − 1.80·11-s + 3.46·12-s − 9.29·15-s − 2.68·20-s + 2.61·21-s + 17/5·25-s − 10.3·27-s + 0.755·28-s − 6.68·29-s − 5.38·31-s + 6.26·33-s − 2.02·35-s − 7·36-s + 1.80·44-s + 18.7·45-s + 49-s − 0.824·53-s − 4.85·55-s + 2.34·59-s + 9.29·60-s − 5.29·63-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.23851\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04837167969\)
\(L(\frac12)\) \(\approx\) \(0.04837167969\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \)
11$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 18 T + 137 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
37$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} ) \)
61$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
79$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} ) \)
89$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
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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

−9.463437604897872984536497197881, −9.377064313900649788573689064210, −9.286755719502426704515534792916, −8.771094644198578161487309569184, −8.511580794288346771858191673026, −7.63798360644569760032079635865, −7.52877949431082005621226028509, −7.17665805796150911591748976546, −7.11434216307483274562168947597, −6.98257942465056893820267452275, −6.08847587932599529403997670380, −6.01063201128961134553312901332, −5.89745629144618398584096662281, −5.58776190811996915368453182123, −5.41072417612169412885880584733, −5.34860888934293696943476691805, −5.06325435010082705782695807136, −4.73995491413624673763265631232, −3.93336120093502218669534907081, −3.70829686570243310677140819631, −3.57254652042876278285779701688, −2.14401427075237738812491018014, −1.97590010121328969886745868285, −1.77186158140741299978715041944, −0.18704754528549063127061355071, 0.18704754528549063127061355071, 1.77186158140741299978715041944, 1.97590010121328969886745868285, 2.14401427075237738812491018014, 3.57254652042876278285779701688, 3.70829686570243310677140819631, 3.93336120093502218669534907081, 4.73995491413624673763265631232, 5.06325435010082705782695807136, 5.34860888934293696943476691805, 5.41072417612169412885880584733, 5.58776190811996915368453182123, 5.89745629144618398584096662281, 6.01063201128961134553312901332, 6.08847587932599529403997670380, 6.98257942465056893820267452275, 7.11434216307483274562168947597, 7.17665805796150911591748976546, 7.52877949431082005621226028509, 7.63798360644569760032079635865, 8.511580794288346771858191673026, 8.771094644198578161487309569184, 9.286755719502426704515534792916, 9.377064313900649788573689064210, 9.463437604897872984536497197881

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