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Properties

Label 8-10e8-1.1-c0e4-0-0
Degree $8$
Conductor $100000000$
Sign $1$
Analytic cond. $6.20338\times 10^{-6}$
Root an. cond. $0.223397$
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 − 5-s − 9-s + 10-s − 2·13-s − 2·17-s + 18-s + 2·26-s − 2·29-s + 32-s + 2·34-s + 3·37-s − 2·41-s + 45-s + 4·49-s + 3·53-s + 2·58-s − 2·61-s − 64-s + 2·65-s − 2·73-s − 3·74-s + 2·82-s + 2·85-s + 3·89-s − 90-s − 2·97-s + ⋯
L(s)  = 1  − 2-s − 5-s − 9-s + 10-s − 2·13-s − 2·17-s + 18-s + 2·26-s − 2·29-s + 32-s + 2·34-s + 3·37-s − 2·41-s + 45-s + 4·49-s + 3·53-s + 2·58-s − 2·61-s − 64-s + 2·65-s − 2·73-s − 3·74-s + 2·82-s + 2·85-s + 3·89-s − 90-s − 2·97-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.20338\times 10^{-6}\)
Root analytic conductor: \(0.223397\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05085992259\)
\(L(\frac12)\) \(\approx\) \(0.05085992259\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−10.72134589933074432226622561992, −10.21714847714142870445182399272, −9.806097530092695307679460379295, −9.621676144300964856013990338315, −9.499591714021149766685915326899, −8.900279530223792754793762110576, −8.795965928933043041405356203034, −8.655325615439447738633677550011, −8.524044957078315048199567134733, −7.79299994935581846419386074728, −7.46697379647670625499317743355, −7.44260981790956792459665870802, −7.40735181615435052329967335639, −6.55853129117874277206299100115, −6.52341597470470667371308755766, −5.99243829011274036691812126531, −5.52389601233870343120666487150, −5.32983035558480357806833535857, −4.84001626020136688398360006484, −4.26746565859084420235790164286, −4.14002826679925869875833378269, −3.82709231667603315271129119450, −2.77583471385176013367479650464, −2.66152548206980973020188319867, −2.16566075758153450204676947752, 2.16566075758153450204676947752, 2.66152548206980973020188319867, 2.77583471385176013367479650464, 3.82709231667603315271129119450, 4.14002826679925869875833378269, 4.26746565859084420235790164286, 4.84001626020136688398360006484, 5.32983035558480357806833535857, 5.52389601233870343120666487150, 5.99243829011274036691812126531, 6.52341597470470667371308755766, 6.55853129117874277206299100115, 7.40735181615435052329967335639, 7.44260981790956792459665870802, 7.46697379647670625499317743355, 7.79299994935581846419386074728, 8.524044957078315048199567134733, 8.655325615439447738633677550011, 8.795965928933043041405356203034, 8.900279530223792754793762110576, 9.499591714021149766685915326899, 9.621676144300964856013990338315, 9.806097530092695307679460379295, 10.21714847714142870445182399272, 10.72134589933074432226622561992

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