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Properties

Label 8-4704e4-1.1-c1e4-0-1
Degree $8$
Conductor $4.896\times 10^{14}$
Sign $1$
Analytic cond. $1.99057\times 10^{6}$
Root an. cond. $6.12875$
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  − 4·3-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.99057\times 10^{6}\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.070251093\)
\(L(\frac12)\) \(\approx\) \(5.070251093\)
\(L(\frac{3}{2})\) 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

−5.80723571886015257991531232081, −5.78047344452961394745938914032, −5.58049588781736835015720578080, −5.40887878256111534441085267908, −5.34281441451836908261189229186, −4.98151786933114370954047413230, −4.64110065011423279962003144473, −4.47509904962284871644111927011, −4.45270962744138871239567962757, −4.04686055999708564411203983664, −3.84962012662835461558522245982, −3.83398255245143760233620189017, −3.82962782758116413342268028909, −3.14452394565212414969609438568, −2.88103441463971688015458233568, −2.70978645889823521791844248293, −2.44896895673999016870476931545, −2.12518167294717510604736540783, −1.93970781213998652455604935787, −1.77199919994115836227266757886, −1.68927087074838855042359134166, −0.969579294099863663035776260396, −0.823959690807534179258195463495, −0.66824442193925949920340621981, −0.47738329641229033875052434439, 0.47738329641229033875052434439, 0.66824442193925949920340621981, 0.823959690807534179258195463495, 0.969579294099863663035776260396, 1.68927087074838855042359134166, 1.77199919994115836227266757886, 1.93970781213998652455604935787, 2.12518167294717510604736540783, 2.44896895673999016870476931545, 2.70978645889823521791844248293, 2.88103441463971688015458233568, 3.14452394565212414969609438568, 3.82962782758116413342268028909, 3.83398255245143760233620189017, 3.84962012662835461558522245982, 4.04686055999708564411203983664, 4.45270962744138871239567962757, 4.47509904962284871644111927011, 4.64110065011423279962003144473, 4.98151786933114370954047413230, 5.34281441451836908261189229186, 5.40887878256111534441085267908, 5.58049588781736835015720578080, 5.78047344452961394745938914032, 5.80723571886015257991531232081

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