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Properties

Label 8-847e4-1.1-c1e4-0-16
Degree $8$
Conductor $514675673281$
Sign $1$
Analytic cond. $2092.38$
Root an. cond. $2.60064$
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  − 2·2-s − 2·3-s − 6·5-s + 4·6-s + 4·7-s + 8-s − 7·9-s + 12·10-s − 8·14-s + 12·15-s − 16-s + 3·17-s + 14·18-s − 3·19-s − 8·21-s − 8·23-s − 2·24-s + 8·25-s + 20·27-s − 3·29-s − 24·30-s − 3·31-s + 10·32-s − 6·34-s − 24·35-s − 7·37-s + 6·38-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 2.68·5-s + 1.63·6-s + 1.51·7-s + 0.353·8-s − 7/3·9-s + 3.79·10-s − 2.13·14-s + 3.09·15-s − 1/4·16-s + 0.727·17-s + 3.29·18-s − 0.688·19-s − 1.74·21-s − 1.66·23-s − 0.408·24-s + 8/5·25-s + 3.84·27-s − 0.557·29-s − 4.38·30-s − 0.538·31-s + 1.76·32-s − 1.02·34-s − 4.05·35-s − 1.15·37-s + 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2092.38\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + 7 T^{3} + 13 T^{4} + 7 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$D_{4}$ \( ( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 28 T^{2} + 87 T^{3} + 229 T^{4} + 87 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} + 5 p T^{3} + 153 T^{4} + 5 p^{2} T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 45 T^{2} - 62 T^{3} + 881 T^{4} - 62 p T^{5} + 45 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 47 T^{2} + 36 T^{3} + 919 T^{4} + 36 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 402 p T^{5} + 83 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 62 T^{2} + 9 p T^{3} + 2319 T^{4} + 9 p^{2} T^{5} + 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 100 T^{2} + 299 T^{3} + 4283 T^{4} + 299 p T^{5} + 100 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 767 p T^{5} + 154 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 107 T^{2} + 470 T^{3} + 5491 T^{4} + 470 p T^{5} + 107 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 162 T^{2} + 1449 T^{3} + 10505 T^{4} + 1449 p T^{5} + 162 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 108 T^{2} + 725 T^{3} + 4961 T^{4} + 725 p T^{5} + 108 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 428 T^{2} + 4725 T^{3} + 42353 T^{4} + 4725 p T^{5} + 428 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 238 T^{2} - 2447 T^{3} + 20599 T^{4} - 2447 p T^{5} + 238 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 201 T^{2} + 812 T^{3} + 17469 T^{4} + 812 p T^{5} + 201 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 176 T^{2} - 1649 T^{3} + 20013 T^{4} - 1649 p T^{5} + 176 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 308 T^{2} + 1735 T^{3} + 35911 T^{4} + 1735 p T^{5} + 308 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - T + 96 T^{2} - 1149 T^{3} + 4403 T^{4} - 1149 p T^{5} + 96 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 3419 p T^{5} + 312 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 278 T^{2} + 2415 T^{3} + 30889 T^{4} + 2415 p T^{5} + 278 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.085483001567012996438474083411, −7.75728479819107769148208176488, −7.52014672448748082833340705762, −7.37913033567510169997803571021, −6.67118869823777533892362717774, −6.67069962201095436602918914291, −6.56330346838690495549164030174, −6.19181435430828911568002523918, −5.93192232557974660511712413134, −5.59403569364558953413087243967, −5.34703554749736745200182853202, −5.27869737104396536450515755179, −5.21855493075898261584187403175, −4.56829578396688278216359956486, −4.34844768310214240668559023161, −4.33310857214353015448147249606, −4.10462960776561866249209415153, −3.56343740058973570363628717545, −3.34772315135984384019674885855, −3.18976922117332109925060579875, −2.95790583942726190808930725941, −2.25386997546661851387075342486, −2.11971515153447433490079254800, −1.52999480160990434108956509824, −1.14360306814480964587242875578, 0, 0, 0, 0, 1.14360306814480964587242875578, 1.52999480160990434108956509824, 2.11971515153447433490079254800, 2.25386997546661851387075342486, 2.95790583942726190808930725941, 3.18976922117332109925060579875, 3.34772315135984384019674885855, 3.56343740058973570363628717545, 4.10462960776561866249209415153, 4.33310857214353015448147249606, 4.34844768310214240668559023161, 4.56829578396688278216359956486, 5.21855493075898261584187403175, 5.27869737104396536450515755179, 5.34703554749736745200182853202, 5.59403569364558953413087243967, 5.93192232557974660511712413134, 6.19181435430828911568002523918, 6.56330346838690495549164030174, 6.67069962201095436602918914291, 6.67118869823777533892362717774, 7.37913033567510169997803571021, 7.52014672448748082833340705762, 7.75728479819107769148208176488, 8.085483001567012996438474083411

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