Properties

Degree $8$
Conductor $7.834\times 10^{15}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more about

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: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
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$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 896 T^{2} + 13608 T^{3} + 153858 T^{4} + 13608 p T^{5} + 896 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.79006508398158899589196735747, −5.55111333698586453285324416415, −5.50865443828201507117624402298, −5.41608629921436858138843734300, −5.23747781490725801724066033127, −4.88834880195464557095852231511, −4.68417843644144082043038189682, −4.40896036068621652515219761873, −4.36788285319638928933365559501, −4.09373436144211366717094137132, −4.06208818729085408413804552982, −3.92411821742212657760680762575, −3.71039628285252853138341468707, −3.30717210351103327513355307926, −3.14517695917122491634398859551, −3.09182820842358978527202303002, −2.74392897487219044537449184496, −2.05017372324685968058761233870, −2.04921832936424813394860869877, −2.03618438086599802599884686423, −1.88348477703646407683202705256, −1.45412699182750151296424549210, −1.45349379591543604287854601545, −1.14995050095539128415554611174, −1.06640451220701885007517204677, 0, 0, 0, 0, 1.06640451220701885007517204677, 1.14995050095539128415554611174, 1.45349379591543604287854601545, 1.45412699182750151296424549210, 1.88348477703646407683202705256, 2.03618438086599802599884686423, 2.04921832936424813394860869877, 2.05017372324685968058761233870, 2.74392897487219044537449184496, 3.09182820842358978527202303002, 3.14517695917122491634398859551, 3.30717210351103327513355307926, 3.71039628285252853138341468707, 3.92411821742212657760680762575, 4.06208818729085408413804552982, 4.09373436144211366717094137132, 4.36788285319638928933365559501, 4.40896036068621652515219761873, 4.68417843644144082043038189682, 4.88834880195464557095852231511, 5.23747781490725801724066033127, 5.41608629921436858138843734300, 5.50865443828201507117624402298, 5.55111333698586453285324416415, 5.79006508398158899589196735747

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