Properties

Degree $8$
Conductor $7.834\times 10^{15}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·5-s + 10·9-s − 4·11-s + 8·13-s + 16·15-s − 4·17-s + 8·19-s + 12·23-s + 4·25-s + 20·27-s + 8·31-s − 16·33-s + 32·39-s − 20·41-s + 8·43-s + 40·45-s + 16·47-s − 16·51-s − 16·55-s + 32·57-s + 16·61-s + 32·65-s + 8·67-s + 48·69-s + 12·71-s − 8·73-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 10/3·9-s − 1.20·11-s + 2.21·13-s + 4.13·15-s − 0.970·17-s + 1.83·19-s + 2.50·23-s + 4/5·25-s + 3.84·27-s + 1.43·31-s − 2.78·33-s + 5.12·39-s − 3.12·41-s + 1.21·43-s + 5.96·45-s + 2.33·47-s − 2.24·51-s − 2.15·55-s + 4.23·57-s + 2.04·61-s + 3.96·65-s + 0.977·67-s + 5.77·69-s + 1.42·71-s − 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(62.11057589\)
\(L(\frac12)\) \(\approx\) \(62.11057589\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 98 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 24 T^{2} + 84 T^{3} + 302 T^{4} + 84 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 48 T^{2} - 232 T^{3} + 978 T^{4} - 232 p T^{5} + 48 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 44 T^{2} + 84 T^{3} + 818 T^{4} + 84 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 68 T^{2} - 392 T^{3} + 1926 T^{4} - 392 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 120 T^{2} - 780 T^{3} + 4350 T^{4} - 780 p T^{5} + 120 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 60 T^{2} - 128 T^{3} + 1862 T^{4} - 128 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 84 T^{2} - 616 T^{3} + 3222 T^{4} - 616 p T^{5} + 84 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 220 T^{2} + 1540 T^{3} + 9874 T^{4} + 1540 p T^{5} + 220 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 648 T^{3} + 6710 T^{4} - 648 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 164 T^{2} - 1232 T^{3} + 170 p T^{4} - 1232 p T^{5} + 164 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 140 T^{2} + 128 T^{3} + 9494 T^{4} + 128 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 84 T^{2} + 960 T^{3} + 1350 T^{4} + 960 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 288 T^{2} - 2768 T^{3} + 27282 T^{4} - 2768 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} + 56 T^{3} + 3286 T^{4} + 56 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 248 T^{2} - 2380 T^{3} + 25406 T^{4} - 2380 p T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 256 T^{2} + 1608 T^{3} + 27170 T^{4} + 1608 p T^{5} + 256 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 284 T^{2} - 2768 T^{3} + 31366 T^{4} - 2768 p T^{5} + 284 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 204 T^{2} + 256 T^{3} + 21878 T^{4} + 256 p T^{5} + 204 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 524 T^{2} + 76 p T^{3} + 72658 T^{4} + 76 p^{2} T^{5} + 524 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 896 T^{2} + 13608 T^{3} + 153858 T^{4} + 13608 p T^{5} + 896 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.38695267708514710161032875363, −5.11137462637700636458642321367, −5.05451721450352533665628525183, −5.02185352679032644562611134027, −4.72249229505787548416671696609, −4.36045507567416496418157317132, −4.10732186472889012429019627679, −4.08783829931642598142165915846, −3.97688846144305640247017059653, −3.46737702166552516749889541728, −3.46412126946019179475609358687, −3.41586471188220590481493813226, −3.03637408711076943902771198643, −2.82970686470315613499665908019, −2.61666689501750252261050700911, −2.59605252897496069167722099109, −2.55963408058261167942453417435, −2.01242376277755888637261136670, −1.91060829225128982499513628728, −1.66311325592417088490708806641, −1.55537560428189180123287616461, −1.19578799521226615499487068169, −0.949830681883362304192684578442, −0.74985174541004993109493342577, −0.52441350944287243730775537613, 0.52441350944287243730775537613, 0.74985174541004993109493342577, 0.949830681883362304192684578442, 1.19578799521226615499487068169, 1.55537560428189180123287616461, 1.66311325592417088490708806641, 1.91060829225128982499513628728, 2.01242376277755888637261136670, 2.55963408058261167942453417435, 2.59605252897496069167722099109, 2.61666689501750252261050700911, 2.82970686470315613499665908019, 3.03637408711076943902771198643, 3.41586471188220590481493813226, 3.46412126946019179475609358687, 3.46737702166552516749889541728, 3.97688846144305640247017059653, 4.08783829931642598142165915846, 4.10732186472889012429019627679, 4.36045507567416496418157317132, 4.72249229505787548416671696609, 5.02185352679032644562611134027, 5.05451721450352533665628525183, 5.11137462637700636458642321367, 5.38695267708514710161032875363

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