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Properties

Label 8-600e4-1.1-c0e4-0-0
Degree $8$
Conductor $129600000000$
Sign $1$
Analytic cond. $0.00803958$
Root an. cond. $0.547210$
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  − 2-s − 3-s − 5-s + 6-s − 2·7-s + 10-s + 3·11-s + 2·14-s + 15-s + 2·21-s − 3·22-s − 2·29-s − 30-s + 3·31-s + 32-s − 3·33-s + 2·35-s − 2·42-s + 49-s − 2·53-s − 3·55-s + 2·58-s + 3·59-s − 3·62-s − 64-s + 3·66-s − 2·70-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s − 2·7-s + 10-s + 3·11-s + 2·14-s + 15-s + 2·21-s − 3·22-s − 2·29-s − 30-s + 3·31-s + 32-s − 3·33-s + 2·35-s − 2·42-s + 49-s − 2·53-s − 3·55-s + 2·58-s + 3·59-s − 3·62-s − 64-s + 3·66-s − 2·70-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.00803958\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1284279928\)
\(L(\frac12)\) \(\approx\) \(0.1284279928\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−7.79572856718561557423301554188, −7.79478188794710945819909703724, −7.32587980236027020902303822767, −7.31852904345565715371835095778, −6.79037239255847860279440445983, −6.68869211954584936440997489726, −6.57981879606240439600326884385, −6.22063280972913537765628957527, −6.12378930611367341329444425505, −5.94778937317783289954358582731, −5.82919538789292654622470160386, −5.17359885429417921345660262352, −4.95105270210484386958247208447, −4.76479160159235241548278419525, −4.21200835832611039258921885783, −4.16739019015564403310701070665, −3.88265714929399562876545015626, −3.62819773786799494361511988085, −3.31534862753941107268371344393, −3.10719051896162680976184441992, −2.71358618425737609668826823728, −2.20860206905284398658715132605, −1.67024574908609273310209588419, −1.16243574400570910889055909247, −0.61723419447780696887501190023, 0.61723419447780696887501190023, 1.16243574400570910889055909247, 1.67024574908609273310209588419, 2.20860206905284398658715132605, 2.71358618425737609668826823728, 3.10719051896162680976184441992, 3.31534862753941107268371344393, 3.62819773786799494361511988085, 3.88265714929399562876545015626, 4.16739019015564403310701070665, 4.21200835832611039258921885783, 4.76479160159235241548278419525, 4.95105270210484386958247208447, 5.17359885429417921345660262352, 5.82919538789292654622470160386, 5.94778937317783289954358582731, 6.12378930611367341329444425505, 6.22063280972913537765628957527, 6.57981879606240439600326884385, 6.68869211954584936440997489726, 6.79037239255847860279440445983, 7.31852904345565715371835095778, 7.32587980236027020902303822767, 7.79478188794710945819909703724, 7.79572856718561557423301554188

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