Properties

Label 8-980e4-1.1-c1e4-0-19
Degree $8$
Conductor $922368160000$
Sign $1$
Analytic cond. $3749.83$
Root an. cond. $2.79738$
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  + 4·2-s + 8·4-s + 8·8-s + 3·9-s + 6·11-s − 4·16-s − 12·17-s + 12·18-s − 12·19-s + 24·22-s + 12·23-s + 25-s + 4·29-s + 12·31-s − 32·32-s − 48·34-s + 24·36-s + 12·37-s − 48·38-s + 48·44-s + 48·46-s + 4·50-s − 4·53-s + 16·58-s − 12·61-s + 48·62-s − 64·64-s + ⋯
L(s)  = 1  + 2.82·2-s + 4·4-s + 2.82·8-s + 9-s + 1.80·11-s − 16-s − 2.91·17-s + 2.82·18-s − 2.75·19-s + 5.11·22-s + 2.50·23-s + 1/5·25-s + 0.742·29-s + 2.15·31-s − 5.65·32-s − 8.23·34-s + 4·36-s + 1.97·37-s − 7.78·38-s + 7.23·44-s + 7.07·46-s + 0.565·50-s − 0.549·53-s + 2.10·58-s − 1.53·61-s + 6.09·62-s − 8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3749.83\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.60550018\)
\(L(\frac12)\) \(\approx\) \(17.60550018\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 - 6 T + 3 p T^{2} - 126 T^{3} + 452 T^{4} - 126 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$D_4\times C_2$ \( 1 + 12 T + 5 p T^{2} + 444 T^{3} + 1896 T^{4} + 444 p T^{5} + 5 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 102 T^{2} - 648 T^{3} + 3491 T^{4} - 648 p T^{5} + 102 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 12 T + 46 T^{2} - 288 T^{3} + 2907 T^{4} - 288 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 91 T^{2} + 6072 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T + 146 T^{2} + 1176 T^{3} + 8811 T^{4} + 1176 p T^{5} + 146 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
71$D_4\times C_2$ \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 24 T + 350 T^{2} - 3792 T^{3} + 33651 T^{4} - 3792 p T^{5} + 350 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 30 T + 497 T^{2} - 5910 T^{3} + 56268 T^{4} - 5910 p T^{5} + 497 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.74171147090917377977860994370, −6.71774929731430437144614805381, −6.67903022590382552243469943456, −6.46379809129479968512335648947, −6.36396626451616348234512030712, −6.18643508280406587908851345962, −5.93098360529568696819004392939, −5.28850408120451442833713215966, −4.98625893177941722753975029297, −4.98616325594771114546157963062, −4.94618407734096966102929401000, −4.44656886547249388875360380107, −4.41197326934266195902820560687, −4.19315761494680560745921142235, −3.93405224305699667806806804212, −3.84286145183779699013496559683, −3.41462033011665597323626924192, −3.21305997107279629833536641923, −2.76948324008645233937316384992, −2.29698675344963884440103051030, −2.29097630985685841803637222810, −2.26795738237768940057691723873, −1.61755160527248034522154097864, −0.943521721056392514703129574114, −0.62838603535870315000632625848, 0.62838603535870315000632625848, 0.943521721056392514703129574114, 1.61755160527248034522154097864, 2.26795738237768940057691723873, 2.29097630985685841803637222810, 2.29698675344963884440103051030, 2.76948324008645233937316384992, 3.21305997107279629833536641923, 3.41462033011665597323626924192, 3.84286145183779699013496559683, 3.93405224305699667806806804212, 4.19315761494680560745921142235, 4.41197326934266195902820560687, 4.44656886547249388875360380107, 4.94618407734096966102929401000, 4.98616325594771114546157963062, 4.98625893177941722753975029297, 5.28850408120451442833713215966, 5.93098360529568696819004392939, 6.18643508280406587908851345962, 6.36396626451616348234512030712, 6.46379809129479968512335648947, 6.67903022590382552243469943456, 6.71774929731430437144614805381, 6.74171147090917377977860994370

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