Properties

Label 8-40e4-1.1-c1e4-0-0
Degree $8$
Conductor $2560000$
Sign $1$
Analytic cond. $0.0104075$
Root an. cond. $0.565156$
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  − 2·2-s + 2·4-s − 4·7-s − 4·8-s + 4·9-s + 8·14-s + 8·16-s − 8·18-s − 4·23-s − 2·25-s − 8·28-s − 8·31-s − 8·32-s + 8·36-s − 8·41-s + 8·46-s + 20·47-s − 12·49-s + 4·50-s + 16·56-s + 16·62-s − 16·63-s + 8·64-s + 8·71-s − 16·72-s + 16·73-s − 32·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 1.41·8-s + 4/3·9-s + 2.13·14-s + 2·16-s − 1.88·18-s − 0.834·23-s − 2/5·25-s − 1.51·28-s − 1.43·31-s − 1.41·32-s + 4/3·36-s − 1.24·41-s + 1.17·46-s + 2.91·47-s − 1.71·49-s + 0.565·50-s + 2.13·56-s + 2.03·62-s − 2.01·63-s + 64-s + 0.949·71-s − 1.88·72-s + 1.87·73-s − 3.60·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.0104075\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2560000,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1803202083\)
\(L(\frac12)\) \(\approx\) \(0.1803202083\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−12.26929008658477693804126621085, −12.10058734442960941037406833041, −11.40092958554935680105153880476, −11.06649261557972531905603987407, −11.03942142649809864668539328989, −10.20076701750556844782565606349, −10.11854950467773633428324413242, −9.877369627672556812380126500789, −9.759763384450441618832338961039, −9.130388608674758292080000760602, −9.128918287191284636320662147749, −8.658063991760145531382247833125, −8.116337803120132320517287048108, −8.003458413310215123129012497393, −7.22925099472382298278932711966, −7.14791678296403584189372620437, −6.83400993459043780951836495240, −6.13409915347686679763897898118, −6.03295757928068588483956934399, −5.53425094987208521971798192883, −4.78864208392702509396680983103, −4.07608247633227766546385885342, −3.55743468534700640706687531154, −3.07834236443164268480890653644, −1.98800828619583758752930483423, 1.98800828619583758752930483423, 3.07834236443164268480890653644, 3.55743468534700640706687531154, 4.07608247633227766546385885342, 4.78864208392702509396680983103, 5.53425094987208521971798192883, 6.03295757928068588483956934399, 6.13409915347686679763897898118, 6.83400993459043780951836495240, 7.14791678296403584189372620437, 7.22925099472382298278932711966, 8.003458413310215123129012497393, 8.116337803120132320517287048108, 8.658063991760145531382247833125, 9.128918287191284636320662147749, 9.130388608674758292080000760602, 9.759763384450441618832338961039, 9.877369627672556812380126500789, 10.11854950467773633428324413242, 10.20076701750556844782565606349, 11.03942142649809864668539328989, 11.06649261557972531905603987407, 11.40092958554935680105153880476, 12.10058734442960941037406833041, 12.26929008658477693804126621085

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