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Properties

Label 8-431e4-1.1-c1e4-0-0
Degree $8$
Conductor $34507149121$
Sign $1$
Analytic cond. $140.287$
Root an. cond. $1.85514$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 4·4-s − 5·5-s + 3·6-s + 2·7-s + 4·8-s − 3·9-s + 5·10-s + 11-s + 12·12-s − 5·13-s − 2·14-s + 15·15-s + 8·16-s + 2·17-s + 3·18-s − 6·19-s + 20·20-s − 6·21-s − 22-s + 4·23-s − 12·24-s − 25-s + 5·26-s + 22·27-s − 8·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 2·4-s − 2.23·5-s + 1.22·6-s + 0.755·7-s + 1.41·8-s − 9-s + 1.58·10-s + 0.301·11-s + 3.46·12-s − 1.38·13-s − 0.534·14-s + 3.87·15-s + 2·16-s + 0.485·17-s + 0.707·18-s − 1.37·19-s + 4.47·20-s − 1.30·21-s − 0.213·22-s + 0.834·23-s − 2.44·24-s − 1/5·25-s + 0.980·26-s + 4.23·27-s − 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(431^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(431^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(431^{4}\)
Sign: $1$
Analytic conductor: \(140.287\)
Root analytic conductor: \(1.85514\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 431^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad431$C_1$ \( ( 1 + T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 5 T^{2} + 5 T^{3} + 13 T^{4} + 5 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 + p T + 4 p T^{2} + 23 T^{3} + 53 T^{4} + 23 p T^{5} + 4 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + p T + 26 T^{2} + 3 p^{2} T^{3} + 209 T^{4} + 3 p^{3} T^{5} + 26 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 26 T^{2} - 39 T^{3} + 267 T^{4} - 39 p T^{5} + 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - T + 31 T^{2} - 2 T^{3} + 421 T^{4} - 2 p T^{5} + 31 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 51 T^{2} + 190 T^{3} + 987 T^{4} + 190 p T^{5} + 51 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 26 T^{2} - 129 T^{3} + 407 T^{4} - 129 p T^{5} + 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 73 T^{2} + 286 T^{3} + 2003 T^{4} + 286 p T^{5} + 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 74 T^{2} - 272 T^{3} + 2347 T^{4} - 272 p T^{5} + 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 96 T^{2} + 15 p T^{3} + 3911 T^{4} + 15 p^{2} T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 355 T^{2} + 3260 T^{3} + 21487 T^{4} + 3260 p T^{5} + 355 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + T + 68 T^{2} + 99 T^{3} + 2403 T^{4} + 99 p T^{5} + 68 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 48 T^{2} + 396 T^{3} + 65 T^{4} + 396 p T^{5} + 48 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 249 T^{2} + 2168 T^{3} + 17697 T^{4} + 2168 p T^{5} + 249 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 99 T^{2} + 278 T^{3} + 4719 T^{4} + 278 p T^{5} + 99 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 109 T^{2} - 632 T^{3} + 7237 T^{4} - 632 p T^{5} + 109 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 210 T^{2} - 795 T^{3} + 17897 T^{4} - 795 p T^{5} + 210 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 134 T^{2} - 400 T^{3} + 8631 T^{4} - 400 p T^{5} + 134 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 220 T^{2} - 1495 T^{3} + 21443 T^{4} - 1495 p T^{5} + 220 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 228 T^{2} - 2050 T^{3} + 22598 T^{4} - 2050 p T^{5} + 228 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 50 T + 1226 T^{2} + 18675 T^{3} + 192207 T^{4} + 18675 p T^{5} + 1226 p^{2} T^{6} + 50 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 120 T^{2} - 823 T^{3} + 13689 T^{4} - 823 p T^{5} + 120 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 316 T^{2} + 3255 T^{3} + 38337 T^{4} + 3255 p T^{5} + 316 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 35 T + 742 T^{2} + 10465 T^{3} + 113793 T^{4} + 10465 p T^{5} + 742 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 228 T^{2} + 1239 T^{3} + 31773 T^{4} + 1239 p T^{5} + 228 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.757064292021243105963585426955, −8.073774787390963978203906137892, −8.066476911164800109232868018766, −8.020210797461077444404711404657, −7.963263487917183300703897059123, −7.28963224504557004120035047656, −7.21911881506331381433316710478, −6.90461449994335567980211015730, −6.77970772386872871752443735318, −6.17237951805918348387135627855, −5.86277632993606046047372893388, −5.72499152615627113324529087330, −5.54880600819072794136180703206, −5.14488017911330079642349793194, −5.02998574093495358985413359712, −4.91299907632159884307848229091, −4.46142114479030989301725855667, −4.15070823578464838448607528747, −3.95148272286541562204111113763, −3.60365302614612384240579114748, −3.51790462767506543012504496676, −2.93182774161174731444984495539, −2.61126313986956073596968394846, −1.68629869110012463569112793413, −1.57497582503403003640892917893, 0, 0, 0, 0, 1.57497582503403003640892917893, 1.68629869110012463569112793413, 2.61126313986956073596968394846, 2.93182774161174731444984495539, 3.51790462767506543012504496676, 3.60365302614612384240579114748, 3.95148272286541562204111113763, 4.15070823578464838448607528747, 4.46142114479030989301725855667, 4.91299907632159884307848229091, 5.02998574093495358985413359712, 5.14488017911330079642349793194, 5.54880600819072794136180703206, 5.72499152615627113324529087330, 5.86277632993606046047372893388, 6.17237951805918348387135627855, 6.77970772386872871752443735318, 6.90461449994335567980211015730, 7.21911881506331381433316710478, 7.28963224504557004120035047656, 7.963263487917183300703897059123, 8.020210797461077444404711404657, 8.066476911164800109232868018766, 8.073774787390963978203906137892, 8.757064292021243105963585426955

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