Properties

Label 8-3528e4-1.1-c0e4-0-15
Degree $8$
Conductor $1.549\times 10^{14}$
Sign $1$
Analytic cond. $9.61042$
Root an. cond. $1.32691$
Motivic weight $0$
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  + 4·3-s − 2·4-s + 10·9-s + 2·11-s − 8·12-s + 3·16-s − 2·17-s + 2·19-s − 25-s + 20·27-s + 8·33-s − 20·36-s − 2·41-s + 2·43-s − 4·44-s + 12·48-s − 8·51-s + 8·57-s − 4·64-s + 4·68-s + 2·73-s − 4·75-s − 4·76-s + 35·81-s − 2·83-s + 2·89-s + 2·97-s + ⋯
L(s)  = 1  + 4·3-s − 2·4-s + 10·9-s + 2·11-s − 8·12-s + 3·16-s − 2·17-s + 2·19-s − 25-s + 20·27-s + 8·33-s − 20·36-s − 2·41-s + 2·43-s − 4·44-s + 12·48-s − 8·51-s + 8·57-s − 4·64-s + 4·68-s + 2·73-s − 4·75-s − 4·76-s + 35·81-s − 2·83-s + 2·89-s + 2·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(9.61042\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.406661669\)
\(L(\frac12)\) \(\approx\) \(8.406661669\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good5$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
13$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
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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

−6.22507845008048094912120490269, −6.18874053534362900128818851912, −6.11728790104566088773916390454, −5.37650265246946683570207605294, −5.21814877617517081970210355329, −5.20070930955735854441343998237, −4.95454209460848301146379348683, −4.59278107555452153253063369024, −4.44605186805871331387859412872, −4.25736269662469275128700646031, −4.20557831668578490276161422937, −3.89974711375387114705896124915, −3.71553242211967050102927765429, −3.54587914931911426509636225280, −3.47396956530080476909438730049, −3.29448408373054880228499096418, −2.83524444749774969684142421509, −2.82480991154033443372409468280, −2.43918535839094800802980463391, −2.05318900107679578454500408818, −1.99308451640444034374467753981, −1.76290038859007705313683713133, −1.21448026492260433858977212928, −1.05296749719297495937178609394, −1.03841722139716003820846220281, 1.03841722139716003820846220281, 1.05296749719297495937178609394, 1.21448026492260433858977212928, 1.76290038859007705313683713133, 1.99308451640444034374467753981, 2.05318900107679578454500408818, 2.43918535839094800802980463391, 2.82480991154033443372409468280, 2.83524444749774969684142421509, 3.29448408373054880228499096418, 3.47396956530080476909438730049, 3.54587914931911426509636225280, 3.71553242211967050102927765429, 3.89974711375387114705896124915, 4.20557831668578490276161422937, 4.25736269662469275128700646031, 4.44605186805871331387859412872, 4.59278107555452153253063369024, 4.95454209460848301146379348683, 5.20070930955735854441343998237, 5.21814877617517081970210355329, 5.37650265246946683570207605294, 6.11728790104566088773916390454, 6.18874053534362900128818851912, 6.22507845008048094912120490269

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