Properties

Label 8-1960e4-1.1-c1e4-0-8
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
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  − 2·3-s − 4·5-s − 9-s + 2·11-s − 10·13-s + 8·15-s − 6·17-s − 4·23-s + 10·25-s + 2·27-s − 2·29-s + 12·31-s − 4·33-s + 20·39-s − 12·41-s − 8·43-s + 4·45-s + 2·47-s + 12·51-s − 4·53-s − 8·55-s − 8·59-s − 20·61-s + 40·65-s − 8·67-s + 8·69-s + 4·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 1/3·9-s + 0.603·11-s − 2.77·13-s + 2.06·15-s − 1.45·17-s − 0.834·23-s + 2·25-s + 0.384·27-s − 0.371·29-s + 2.15·31-s − 0.696·33-s + 3.20·39-s − 1.87·41-s − 1.21·43-s + 0.596·45-s + 0.291·47-s + 1.68·51-s − 0.549·53-s − 1.07·55-s − 1.04·59-s − 2.56·61-s + 4.96·65-s − 0.977·67-s + 0.963·69-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 26 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 3 p T^{2} - 46 T^{3} + 480 T^{4} - 46 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 71 T^{2} + 2 p^{2} T^{3} + 1384 T^{4} + 2 p^{3} T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + p T^{2} + 62 T^{3} + 434 T^{4} + 62 p T^{5} + p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} - 24 T^{3} + 1186 T^{4} - 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 54 T^{2} + 324 T^{3} + 1442 T^{4} + 324 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 134 T^{2} - 828 T^{3} + 5602 T^{4} - 828 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 106 T^{2} - 40 T^{3} + 5114 T^{4} - 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 68 T^{2} - 52 T^{3} - 2014 T^{4} - 52 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 124 T^{2} + 840 T^{3} + 7718 T^{4} + 840 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 147 T^{2} - 378 T^{3} + 9344 T^{4} - 378 p T^{5} + 147 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + 8866 T^{4} + 4 p^{2} T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 288 T^{2} + 2572 T^{3} + 22350 T^{4} + 2572 p T^{5} + 288 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 154 T^{2} + 1176 T^{3} + 13994 T^{4} + 1176 p T^{5} + 154 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 12 p T^{3} + 5438 T^{4} - 12 p^{2} T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 122 T^{2} - 240 T^{3} - 7038 T^{4} - 240 p T^{5} + 122 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 389 T^{2} - 4726 T^{3} + 48924 T^{4} - 4726 p T^{5} + 389 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 36 T + 750 T^{2} + 10564 T^{3} + 110978 T^{4} + 10564 p T^{5} + 750 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 914 T^{2} + 13800 T^{3} + 152258 T^{4} + 13800 p T^{5} + 914 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 545 T^{2} + 7602 T^{3} + 85890 T^{4} + 7602 p T^{5} + 545 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.87730835825396642652039924220, −6.61914297810571913650687893546, −6.59967966553750602937869631105, −6.43694501808184767494419895831, −6.26988213219476380348217427456, −5.69899876905843456346627494237, −5.67721707064155464214334160785, −5.42913181580361057719299083629, −5.38837526219137806507359950458, −4.91873693141869133855208523497, −4.68949748462580548840060012322, −4.62236045265392216322657400640, −4.45664738273294100826276258138, −4.28754350583900050823602510376, −3.98309460498109388316750784547, −3.78401155792956527053487969812, −3.53284930930702936662707404025, −2.84841699672278230401022538293, −2.83575483568382191495233126475, −2.80108467574215255561659206745, −2.75883715249086266604990613246, −1.95183775920225035551493518325, −1.83096612639558045798135573150, −1.29316677628155715891351753069, −1.21486236431899542509165270072, 0, 0, 0, 0, 1.21486236431899542509165270072, 1.29316677628155715891351753069, 1.83096612639558045798135573150, 1.95183775920225035551493518325, 2.75883715249086266604990613246, 2.80108467574215255561659206745, 2.83575483568382191495233126475, 2.84841699672278230401022538293, 3.53284930930702936662707404025, 3.78401155792956527053487969812, 3.98309460498109388316750784547, 4.28754350583900050823602510376, 4.45664738273294100826276258138, 4.62236045265392216322657400640, 4.68949748462580548840060012322, 4.91873693141869133855208523497, 5.38837526219137806507359950458, 5.42913181580361057719299083629, 5.67721707064155464214334160785, 5.69899876905843456346627494237, 6.26988213219476380348217427456, 6.43694501808184767494419895831, 6.59967966553750602937869631105, 6.61914297810571913650687893546, 6.87730835825396642652039924220

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