Properties

Label 8-1960e4-1.1-c1e4-0-5
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
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  − 3-s − 2·5-s + 2·9-s + 11-s + 2·13-s + 2·15-s − 11·17-s + 6·19-s + 2·23-s + 25-s + 27-s + 10·29-s + 4·31-s − 33-s − 2·39-s + 12·41-s − 12·43-s − 4·45-s − 9·47-s + 11·51-s + 18·53-s − 2·55-s − 6·57-s + 8·59-s + 22·61-s − 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 2/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s − 2.66·17-s + 1.37·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.174·33-s − 0.320·39-s + 1.87·41-s − 1.82·43-s − 0.596·45-s − 1.31·47-s + 1.54·51-s + 2.47·53-s − 0.269·55-s − 0.794·57-s + 1.04·59-s + 2.81·61-s − 0.496·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.820256731\)
\(L(\frac12)\) \(\approx\) \(3.820256731\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + T - T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - T - 17 T^{2} + 4 T^{3} + 192 T^{4} + 4 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 11 T + 61 T^{2} + 286 T^{3} + 1254 T^{4} + 286 p T^{5} + 61 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 6 T + 6 T^{2} + 48 T^{3} - 145 T^{4} + 48 p T^{5} + 6 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 2 T - 26 T^{2} + 32 T^{3} + 279 T^{4} + 32 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 4 T + 18 T^{2} + 256 T^{3} - 1453 T^{4} + 256 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 - 6 T^{2} - 1333 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 9 T - 29 T^{2} + 144 T^{3} + 6084 T^{4} + 144 p T^{5} - 29 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 18 T + 154 T^{2} - 1152 T^{3} + 8919 T^{4} - 1152 p T^{5} + 154 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 22 T + 258 T^{2} - 2288 T^{3} + 17831 T^{4} - 2288 p T^{5} + 258 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 12 T + 42 T^{2} + 384 T^{3} - 3733 T^{4} + 384 p T^{5} + 42 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 8 T - 30 T^{2} - 416 T^{3} - 1165 T^{4} - 416 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 11 T - 29 T^{2} + 88 T^{3} + 6700 T^{4} + 88 p T^{5} - 29 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
89$D_4\times C_2$ \( 1 + 2 T - 158 T^{2} - 32 T^{3} + 17967 T^{4} - 32 p T^{5} - 158 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.70314395505098318796910876334, −6.38620415015256850602101468110, −6.22427595404253358643510150780, −5.98979843877545560268487148543, −5.73039350248098011870815815232, −5.21492397142678474931529905655, −5.11923213507100826667271196293, −5.08260023715118558597720071206, −4.99257432843586446438619535891, −4.56529238409860145833987799455, −4.32853940226450346406140532239, −4.10062904976725295696695376447, −4.06245689243471321473477668390, −3.68965768926719339848888064060, −3.39253223758680851251209378741, −3.37416680890955103755450107589, −2.95009555542848352465098475576, −2.53521189925904769209730740114, −2.37110589791773745211642624128, −2.12340510956049156368630618701, −1.93315291020927477133867197303, −1.32428535768236096331549338380, −0.842665024648229601038197617345, −0.78308436217572703223338157871, −0.52685535124996996378738375750, 0.52685535124996996378738375750, 0.78308436217572703223338157871, 0.842665024648229601038197617345, 1.32428535768236096331549338380, 1.93315291020927477133867197303, 2.12340510956049156368630618701, 2.37110589791773745211642624128, 2.53521189925904769209730740114, 2.95009555542848352465098475576, 3.37416680890955103755450107589, 3.39253223758680851251209378741, 3.68965768926719339848888064060, 4.06245689243471321473477668390, 4.10062904976725295696695376447, 4.32853940226450346406140532239, 4.56529238409860145833987799455, 4.99257432843586446438619535891, 5.08260023715118558597720071206, 5.11923213507100826667271196293, 5.21492397142678474931529905655, 5.73039350248098011870815815232, 5.98979843877545560268487148543, 6.22427595404253358643510150780, 6.38620415015256850602101468110, 6.70314395505098318796910876334

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