Properties

Label 8-570e4-1.1-c1e4-0-0
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $429.148$
Root an. cond. $2.13341$
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 − 2·3-s + 4-s + 2·5-s − 4·6-s − 2·8-s + 9-s + 4·10-s − 12·11-s − 2·12-s + 8·13-s − 4·15-s − 4·16-s + 2·17-s + 2·18-s + 2·20-s − 24·22-s + 6·23-s + 4·24-s + 25-s + 16·26-s + 2·27-s − 10·29-s − 8·30-s − 8·31-s − 2·32-s + 24·33-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s + 1.26·10-s − 3.61·11-s − 0.577·12-s + 2.21·13-s − 1.03·15-s − 16-s + 0.485·17-s + 0.471·18-s + 0.447·20-s − 5.11·22-s + 1.25·23-s + 0.816·24-s + 1/5·25-s + 3.13·26-s + 0.384·27-s − 1.85·29-s − 1.46·30-s − 1.43·31-s − 0.353·32-s + 4.17·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(429.148\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1181515147\)
\(L(\frac12)\) \(\approx\) \(0.1181515147\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
good7$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 2 T - 12 T^{2} + 36 T^{3} - 101 T^{4} + 36 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 10 T + 36 T^{2} + 60 T^{3} + 355 T^{4} + 60 p T^{5} + 36 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 4 T - 51 T^{2} + 60 T^{3} + 2152 T^{4} + 60 p T^{5} - 51 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 65 T^{2} + 1416 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 8 T + 6 T^{2} - 480 T^{3} - 3893 T^{4} - 480 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 2 T - 100 T^{2} - 36 T^{3} + 6851 T^{4} - 36 p T^{5} - 100 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 14 T + 32 T^{2} + 420 T^{3} + 9275 T^{4} + 420 p T^{5} + 32 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 6 T + 56 T^{2} - 972 T^{3} - 7197 T^{4} - 972 p T^{5} + 56 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 18 T + 116 T^{2} + 1116 T^{3} + 14283 T^{4} + 1116 p T^{5} + 116 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 8 T - 34 T^{2} - 480 T^{3} - 1693 T^{4} - 480 p T^{5} - 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
89$D_4\times C_2$ \( 1 - 8 T - 111 T^{2} + 24 T^{3} + 17008 T^{4} + 24 p T^{5} - 111 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 2 T - 20 T^{2} + 340 T^{3} - 9221 T^{4} + 340 p T^{5} - 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.48364351778155417051606713270, −7.38241732797961059606230005261, −7.29974219334851439077275392342, −7.01502423702084290258210777783, −6.84818255786111597811264061326, −6.08073476862071474196335228095, −6.07795267019598013796138337880, −5.90690892526157251968392702098, −5.67232768690701720642479908622, −5.51709216339812757143367484910, −5.40540472116662672330317183893, −5.15533378634657451355223621147, −5.07083348932359159497437956352, −4.53924983221258163860377843305, −4.18111194001153863925359685124, −4.12981803448696506781788778313, −3.53901899644976064063881312067, −3.49853706084626224085784772319, −3.05096724082480074397578310433, −2.65727370149076214949415935096, −2.62528540222150442095753642479, −2.08740078002895903077963268781, −1.50848181995704059546899984299, −1.23427886987856013288839635268, −0.089032823645111286563519576607, 0.089032823645111286563519576607, 1.23427886987856013288839635268, 1.50848181995704059546899984299, 2.08740078002895903077963268781, 2.62528540222150442095753642479, 2.65727370149076214949415935096, 3.05096724082480074397578310433, 3.49853706084626224085784772319, 3.53901899644976064063881312067, 4.12981803448696506781788778313, 4.18111194001153863925359685124, 4.53924983221258163860377843305, 5.07083348932359159497437956352, 5.15533378634657451355223621147, 5.40540472116662672330317183893, 5.51709216339812757143367484910, 5.67232768690701720642479908622, 5.90690892526157251968392702098, 6.07795267019598013796138337880, 6.08073476862071474196335228095, 6.84818255786111597811264061326, 7.01502423702084290258210777783, 7.29974219334851439077275392342, 7.38241732797961059606230005261, 7.48364351778155417051606713270

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