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Properties

Label 8-4704e4-1.1-c1e4-0-2
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

\(L(1)\) \(\approx\) \(72.64650290\)
\(L(\frac12)\) \(\approx\) \(72.64650290\)
\(L(\frac{3}{2})\) 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

−6.10163729902125365843986256353, −5.42206006898363675308524729429, −5.40878211144539251603515444114, −5.28775670201575671193204214118, −5.25031010699411657681864936736, −4.91451343226632032761867945045, −4.74244104947422489230276713314, −4.26153808730787182486091464861, −4.25182210017421464562837751458, −3.81242805297552321738360144825, −3.76311890154040227166049881939, −3.62996158578987141425427802680, −3.42993025958501169400738866604, −3.11201977133959435210368152699, −3.04627472666482723141256637447, −2.91263910094936057142436797505, −2.64839904851103242357005260168, −2.13920170265743500435690199180, −1.91816150153738173301086604779, −1.86719160822634474275619884883, −1.82890126691397605728281817187, −1.23360813198237427455226400396, −1.00632302306326863466282982463, −0.924349618637425819444354498474, −0.805241075556060165557432657410, 0.805241075556060165557432657410, 0.924349618637425819444354498474, 1.00632302306326863466282982463, 1.23360813198237427455226400396, 1.82890126691397605728281817187, 1.86719160822634474275619884883, 1.91816150153738173301086604779, 2.13920170265743500435690199180, 2.64839904851103242357005260168, 2.91263910094936057142436797505, 3.04627472666482723141256637447, 3.11201977133959435210368152699, 3.42993025958501169400738866604, 3.62996158578987141425427802680, 3.76311890154040227166049881939, 3.81242805297552321738360144825, 4.25182210017421464562837751458, 4.26153808730787182486091464861, 4.74244104947422489230276713314, 4.91451343226632032761867945045, 5.25031010699411657681864936736, 5.28775670201575671193204214118, 5.40878211144539251603515444114, 5.42206006898363675308524729429, 6.10163729902125365843986256353

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