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Properties

Label 8-5054e4-1.1-c1e4-0-3
Degree $8$
Conductor $6.524\times 10^{14}$
Sign $1$
Analytic cond. $2.65246\times 10^{6}$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 10·4-s + 2·5-s + 16·6-s + 4·7-s − 20·8-s + 3·9-s − 8·10-s + 4·11-s − 40·12-s + 2·13-s − 16·14-s − 8·15-s + 35·16-s + 2·17-s − 12·18-s + 20·20-s − 16·21-s − 16·22-s + 4·23-s + 80·24-s − 10·25-s − 8·26-s + 10·27-s + 40·28-s − 12·29-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 9-s − 2.52·10-s + 1.20·11-s − 11.5·12-s + 0.554·13-s − 4.27·14-s − 2.06·15-s + 35/4·16-s + 0.485·17-s − 2.82·18-s + 4.47·20-s − 3.49·21-s − 3.41·22-s + 0.834·23-s + 16.3·24-s − 2·25-s − 1.56·26-s + 1.92·27-s + 7.55·28-s − 2.22·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(2.65246\times 10^{6}\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
19 \( 1 \)
good3$C_4\times C_2$ \( 1 + 4 T + 13 T^{2} + 10 p T^{3} + 61 T^{4} + 10 p^{2} T^{5} + 13 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 14 T^{2} - 28 T^{3} + 91 T^{4} - 28 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 30 T^{2} - 96 T^{3} + 479 T^{4} - 96 p T^{5} + 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 41 T^{2} - 86 T^{3} + 729 T^{4} - 86 p T^{5} + 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 57 T^{2} - 70 T^{3} + 1341 T^{4} - 70 p T^{5} + 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 78 T^{2} - 240 T^{3} + 2591 T^{4} - 240 p T^{5} + 78 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 150 T^{2} + 1032 T^{3} + 6999 T^{4} + 1032 p T^{5} + 150 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 150 T^{2} + 1096 T^{3} + 7139 T^{4} + 1096 p T^{5} + 150 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 314 T^{2} + 2768 T^{3} + 18939 T^{4} + 2768 p T^{5} + 314 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 69 T^{2} + 310 T^{3} + 61 p T^{4} + 310 p T^{5} + 69 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 203 T^{2} - 1740 T^{3} + 13581 T^{4} - 1740 p T^{5} + 203 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 62 T^{2} - 320 T^{3} + 691 T^{4} - 320 p T^{5} + 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 122 T^{2} - 160 T^{3} + 8219 T^{4} - 160 p T^{5} + 122 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 150 T^{2} + 552 T^{3} + 10859 T^{4} + 552 p T^{5} + 150 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 118 T^{2} - 204 T^{3} + 10355 T^{4} - 204 p T^{5} + 118 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 22 T + 287 T^{2} + 2200 T^{3} + 17361 T^{4} + 2200 p T^{5} + 287 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 120 T^{2} + 36 T^{3} + 6014 T^{4} + 36 p T^{5} + 120 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 177 T^{2} - 1430 T^{3} + 16989 T^{4} - 1430 p T^{5} + 177 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 75 T^{2} + 1132 T^{3} + 1589 T^{4} + 1132 p T^{5} + 75 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 132 T^{2} + 17814 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 241 T^{2} + 2630 T^{3} + 28261 T^{4} + 2630 p T^{5} + 241 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 22 T + 322 T^{2} - 2860 T^{3} + 26911 T^{4} - 2860 p T^{5} + 322 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.39013796786194019764870263737, −5.89670339184423413738771585170, −5.78630033630633827716717907421, −5.69996157972434381085509808231, −5.65057142658573358394037171362, −5.25220196868334774423815526037, −5.20850865405195814071073105962, −5.16601817345037324618661804360, −5.05295582610987117073104588661, −4.53857457425808245355643015855, −3.97033241021148469755500255307, −3.94181568554772876726234223619, −3.89087079605427936094510607211, −3.75302337706197026928393732058, −3.34674359738136491495279992395, −3.02184756902616802639398313725, −2.90173028449849663188531584096, −2.37201876420806294025454522911, −2.14806984491410479924923779701, −2.14521828909898048430111621300, −1.84309456000180082920346313489, −1.40914525504443773539442169916, −1.35953987527200304401341289896, −1.24288302386602278242716964416, −1.05359567507249449626605028299, 0, 0, 0, 0, 1.05359567507249449626605028299, 1.24288302386602278242716964416, 1.35953987527200304401341289896, 1.40914525504443773539442169916, 1.84309456000180082920346313489, 2.14521828909898048430111621300, 2.14806984491410479924923779701, 2.37201876420806294025454522911, 2.90173028449849663188531584096, 3.02184756902616802639398313725, 3.34674359738136491495279992395, 3.75302337706197026928393732058, 3.89087079605427936094510607211, 3.94181568554772876726234223619, 3.97033241021148469755500255307, 4.53857457425808245355643015855, 5.05295582610987117073104588661, 5.16601817345037324618661804360, 5.20850865405195814071073105962, 5.25220196868334774423815526037, 5.65057142658573358394037171362, 5.69996157972434381085509808231, 5.78630033630633827716717907421, 5.89670339184423413738771585170, 6.39013796786194019764870263737

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