Properties

Label 8-287e4-1.1-c1e4-0-1
Degree $8$
Conductor $6784652161$
Sign $1$
Analytic cond. $27.5826$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 3·4-s + 2·5-s + 2·6-s − 8·7-s − 4·8-s + 7·9-s − 2·10-s − 2·11-s − 6·12-s + 8·14-s − 4·15-s + 6·16-s − 4·17-s − 7·18-s + 6·19-s + 6·20-s + 16·21-s + 2·22-s + 4·23-s + 8·24-s + 11·25-s − 22·27-s − 24·28-s + 4·30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 3.02·7-s − 1.41·8-s + 7/3·9-s − 0.632·10-s − 0.603·11-s − 1.73·12-s + 2.13·14-s − 1.03·15-s + 3/2·16-s − 0.970·17-s − 1.64·18-s + 1.37·19-s + 1.34·20-s + 3.49·21-s + 0.426·22-s + 0.834·23-s + 1.63·24-s + 11/5·25-s − 4.23·27-s − 4.53·28-s + 0.730·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{4} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(27.5826\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{4} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6768837549\)
\(L(\frac12)\) \(\approx\) \(0.6768837549\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 + T - p T^{2} - T^{3} + 3 T^{4} - p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2^2$ \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T - p T^{2} - 4 T^{3} + 528 T^{4} - 4 p T^{5} - p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 6 T + 9 T^{2} + 66 T^{3} - 316 T^{4} + 66 p T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 4 T + 11 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} + 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 4 T - 45 T^{2} - 4 T^{3} + 2264 T^{4} - 4 p T^{5} - 45 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 2 T - 51 T^{2} - 38 T^{3} + 1508 T^{4} - 38 p T^{5} - 51 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 2 T - 71 T^{2} - 38 T^{3} + 3228 T^{4} - 38 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 16 T + 91 T^{2} - 944 T^{3} + 10848 T^{4} - 944 p T^{5} + 91 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 4 T - 61 T^{2} - 164 T^{3} + 1504 T^{4} - 164 p T^{5} - 61 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 10 T - 27 T^{2} + 50 T^{3} + 6308 T^{4} + 50 p T^{5} - 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T - 27 T^{2} + 426 T^{3} - 2932 T^{4} + 426 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 14 T + 21 T^{2} - 406 T^{3} + 10988 T^{4} - 406 p T^{5} + 21 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 24 T + 259 T^{2} - 3336 T^{3} + 41304 T^{4} - 3336 p T^{5} + 259 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.762044341983082142054497535317, −8.580423976387687717036542409587, −8.039446688044265073909879385384, −7.49882096107571855039872080087, −7.40398738329075610382587225464, −7.09735130777205936866009321661, −6.93322798760280802768147733316, −6.85215040429378768294383218778, −6.79496532720299156991386605132, −6.08885885001394570420387556522, −5.96609076022523136848228535054, −5.94496804984481610886579400464, −5.73928340159320297056935962984, −5.20757442594158331473375367004, −4.78397126170656020804853730748, −4.71520534815571736023667794479, −4.06654847480931455866256487292, −3.71065248310360537097622664703, −3.15200471474871487295236106604, −3.11995816372698908113105286069, −2.94644447018086974448287859589, −2.07373918354998676449009635912, −1.99729129393781161489128864507, −1.25856922677398735286318870674, −0.49220371385718397235918026160, 0.49220371385718397235918026160, 1.25856922677398735286318870674, 1.99729129393781161489128864507, 2.07373918354998676449009635912, 2.94644447018086974448287859589, 3.11995816372698908113105286069, 3.15200471474871487295236106604, 3.71065248310360537097622664703, 4.06654847480931455866256487292, 4.71520534815571736023667794479, 4.78397126170656020804853730748, 5.20757442594158331473375367004, 5.73928340159320297056935962984, 5.94496804984481610886579400464, 5.96609076022523136848228535054, 6.08885885001394570420387556522, 6.79496532720299156991386605132, 6.85215040429378768294383218778, 6.93322798760280802768147733316, 7.09735130777205936866009321661, 7.40398738329075610382587225464, 7.49882096107571855039872080087, 8.039446688044265073909879385384, 8.580423976387687717036542409587, 8.762044341983082142054497535317

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