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Properties

Label 8-2000e4-1.1-c1e4-0-11
Degree $8$
Conductor $1.600\times 10^{13}$
Sign $1$
Analytic cond. $65047.1$
Root an. cond. $3.99625$
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 + 6·7-s + 5·9-s − 4·13-s − 4·17-s + 24·21-s + 14·23-s + 2·27-s + 10·29-s + 16·31-s − 16·39-s + 2·41-s + 12·43-s + 16·47-s + 5·49-s − 16·51-s − 24·53-s − 8·59-s + 6·61-s + 30·63-s − 16·67-s + 56·69-s + 24·71-s − 4·73-s + 48·79-s + 4·81-s + 2·83-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.26·7-s + 5/3·9-s − 1.10·13-s − 0.970·17-s + 5.23·21-s + 2.91·23-s + 0.384·27-s + 1.85·29-s + 2.87·31-s − 2.56·39-s + 0.312·41-s + 1.82·43-s + 2.33·47-s + 5/7·49-s − 2.24·51-s − 3.29·53-s − 1.04·59-s + 0.768·61-s + 3.77·63-s − 1.95·67-s + 6.74·69-s + 2.84·71-s − 0.468·73-s + 5.40·79-s + 4/9·81-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(65047.1\)
Root analytic conductor: \(3.99625\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.62072576\)
\(L(\frac12)\) \(\approx\) \(20.62072576\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 11 T^{2} - 26 T^{3} + 53 T^{4} - 26 p T^{5} + 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 31 T^{2} - 2 p^{2} T^{3} + 305 T^{4} - 2 p^{3} T^{5} + 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 12 T^{2} + 18 p T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} + 44 T^{3} + 166 T^{4} + 44 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 32 T^{2} + 44 T^{3} + 430 T^{4} + 44 p T^{5} + 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 24 T^{2} + 366 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 135 T^{2} - 954 T^{3} + 5081 T^{4} - 954 p T^{5} + 135 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 63 T^{2} - 60 T^{3} - 67 T^{4} - 60 p T^{5} + 63 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 188 T^{2} - 48 p T^{3} + 9654 T^{4} - 48 p^{2} T^{5} + 188 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 56 T^{2} + 400 T^{3} + 942 T^{4} + 400 p T^{5} + 56 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 51 T^{2} + 368 T^{3} + 69 T^{4} + 368 p T^{5} + 51 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 219 T^{2} - 1614 T^{3} + 15165 T^{4} - 1614 p T^{5} + 219 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 239 T^{2} - 2102 T^{3} + 17509 T^{4} - 2102 p T^{5} + 239 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 188 T^{2} + 1320 T^{3} + 15206 T^{4} + 1320 p T^{5} + 188 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 143 T^{2} - 688 T^{3} + 12309 T^{4} - 688 p T^{5} + 143 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 296 T^{2} + 2928 T^{3} + 30030 T^{4} + 2928 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 408 T^{2} - 4472 T^{3} + 42974 T^{4} - 4472 p T^{5} + 408 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} + 124 T^{3} + 4086 T^{4} + 124 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 1152 T^{2} - 17616 T^{3} + 186414 T^{4} - 17616 p T^{5} + 1152 p^{2} T^{6} - 48 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 219 T^{2} - 634 T^{3} + 22545 T^{4} - 634 p T^{5} + 219 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 219 T^{2} - 2260 T^{3} + 23381 T^{4} - 2260 p T^{5} + 219 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 244 T^{2} + 668 T^{3} + 31894 T^{4} + 668 p T^{5} + 244 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.66118579206232392600930456894, −6.43625093345871499108157574963, −6.21416229752369196635700366816, −5.78116148130477178244357854970, −5.76870027745525971904548255224, −5.08823618260214198772981921615, −5.05209820265737628834473689079, −5.00150700354492865055111846414, −4.74590281482863876866316196357, −4.69633708232513673987618196630, −4.38124155899139069370862905418, −4.11894617280491080814812282340, −4.07457744187510653839500196360, −3.30747257186227308522299180264, −3.17609613161938408156691094661, −3.16182394710794318953144073599, −3.03486604039533207974415379832, −2.46886193048400048852046759666, −2.39496609862064704924461456942, −2.21693885149667966053706826182, −2.18463209499996918381827691362, −1.45060386208861312274167088956, −1.26253263778341758220623994419, −0.826702084415931452642669852165, −0.70180390740474706208017333128, 0.70180390740474706208017333128, 0.826702084415931452642669852165, 1.26253263778341758220623994419, 1.45060386208861312274167088956, 2.18463209499996918381827691362, 2.21693885149667966053706826182, 2.39496609862064704924461456942, 2.46886193048400048852046759666, 3.03486604039533207974415379832, 3.16182394710794318953144073599, 3.17609613161938408156691094661, 3.30747257186227308522299180264, 4.07457744187510653839500196360, 4.11894617280491080814812282340, 4.38124155899139069370862905418, 4.69633708232513673987618196630, 4.74590281482863876866316196357, 5.00150700354492865055111846414, 5.05209820265737628834473689079, 5.08823618260214198772981921615, 5.76870027745525971904548255224, 5.78116148130477178244357854970, 6.21416229752369196635700366816, 6.43625093345871499108157574963, 6.66118579206232392600930456894

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