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Properties

Label 8-8016e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.129\times 10^{15}$
Sign $1$
Analytic cond. $1.67856\times 10^{7}$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 5·5-s − 7-s + 10·9-s + 8·13-s + 20·15-s + 10·17-s + 2·19-s − 4·21-s − 2·23-s + 5·25-s + 20·27-s + 14·29-s − 31-s − 5·35-s + 5·37-s + 32·39-s + 14·41-s − 2·43-s + 50·45-s − 7·47-s − 7·49-s + 40·51-s + 3·53-s + 8·57-s + 5·59-s + 2·61-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.23·5-s − 0.377·7-s + 10/3·9-s + 2.21·13-s + 5.16·15-s + 2.42·17-s + 0.458·19-s − 0.872·21-s − 0.417·23-s + 25-s + 3.84·27-s + 2.59·29-s − 0.179·31-s − 0.845·35-s + 0.821·37-s + 5.12·39-s + 2.18·41-s − 0.304·43-s + 7.45·45-s − 1.02·47-s − 49-s + 5.60·51-s + 0.412·53-s + 1.05·57-s + 0.650·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 167^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 167^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 167^{4}\)
Sign: $1$
Analytic conductor: \(1.67856\times 10^{7}\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 167^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(83.14931981\)
\(L(\frac12)\) \(\approx\) \(83.14931981\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
167$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - p T + 4 p T^{2} - 11 p T^{3} + 142 T^{4} - 11 p^{2} T^{5} + 4 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + T + 8 T^{2} + 3 p T^{3} + 78 T^{4} + 3 p^{2} T^{5} + 8 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2 \wr S_4$ \( 1 - 8 T + 56 T^{2} - 256 T^{3} + 1102 T^{4} - 256 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 10 T + 88 T^{2} - 494 T^{3} + 2398 T^{4} - 494 p T^{5} + 88 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2 T + 44 T^{2} - 82 T^{3} + 1078 T^{4} - 82 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 2 T + 60 T^{2} + 106 T^{3} + 1830 T^{4} + 106 p T^{5} + 60 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 14 T + 152 T^{2} - 1162 T^{3} + 7102 T^{4} - 1162 p T^{5} + 152 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + T + 88 T^{2} - 3 T^{3} + 3470 T^{4} - 3 p T^{5} + 88 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 5 T + 82 T^{2} - 371 T^{3} + 3898 T^{4} - 371 p T^{5} + 82 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 14 T + 152 T^{2} - 1242 T^{3} + 8030 T^{4} - 1242 p T^{5} + 152 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 2 T + 140 T^{2} + 298 T^{3} + 8374 T^{4} + 298 p T^{5} + 140 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 7 T + 156 T^{2} + 779 T^{3} + 10182 T^{4} + 779 p T^{5} + 156 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 3 T + 172 T^{2} - 441 T^{3} + 12622 T^{4} - 441 p T^{5} + 172 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 5 T + 120 T^{2} - 921 T^{3} + 8006 T^{4} - 921 p T^{5} + 120 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 2 T + 208 T^{2} - 294 T^{3} + 18062 T^{4} - 294 p T^{5} + 208 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 7 T + 82 T^{2} + 441 T^{3} - 1198 T^{4} + 441 p T^{5} + 82 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 116 T^{2} + 298 T^{3} + 13174 T^{4} + 298 p T^{5} + 116 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2 T + 160 T^{2} - 366 T^{3} + 13214 T^{4} - 366 p T^{5} + 160 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 10 T + 260 T^{2} - 1994 T^{3} + 30070 T^{4} - 1994 p T^{5} + 260 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 13 T + 236 T^{2} + 2465 T^{3} + 29470 T^{4} + 2465 p T^{5} + 236 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 13 T + 354 T^{2} - 2955 T^{3} + 45722 T^{4} - 2955 p T^{5} + 354 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 5 T + 242 T^{2} - 827 T^{3} + 31386 T^{4} - 827 p T^{5} + 242 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.69553977292108452936618179775, −5.20849405937760450944233503296, −5.06820664799839549207087459323, −5.06408587302623030953624558931, −4.89660755379957055463950842754, −4.30939205241149376849104076150, −4.25249171471864487990168816978, −4.20463452505005421532302586334, −4.01610641620631014532253822670, −3.55492863029438337318198097578, −3.45651291624323972820686779589, −3.42585427851412239403730044176, −3.28167921675390377223356960501, −2.83657092399691397297121727945, −2.80804388058395370196172593860, −2.64371969057142825289568650426, −2.48335168892101473527837465214, −1.96185677486504667243065855093, −1.91429545866810556926623857323, −1.76242458560598234595187591527, −1.63085657376649837564411545998, −1.18316245138188585173662989672, −0.937569289000548086375443105324, −0.851341147501869465971857715851, −0.63255404388318323404800543044, 0.63255404388318323404800543044, 0.851341147501869465971857715851, 0.937569289000548086375443105324, 1.18316245138188585173662989672, 1.63085657376649837564411545998, 1.76242458560598234595187591527, 1.91429545866810556926623857323, 1.96185677486504667243065855093, 2.48335168892101473527837465214, 2.64371969057142825289568650426, 2.80804388058395370196172593860, 2.83657092399691397297121727945, 3.28167921675390377223356960501, 3.42585427851412239403730044176, 3.45651291624323972820686779589, 3.55492863029438337318198097578, 4.01610641620631014532253822670, 4.20463452505005421532302586334, 4.25249171471864487990168816978, 4.30939205241149376849104076150, 4.89660755379957055463950842754, 5.06408587302623030953624558931, 5.06820664799839549207087459323, 5.20849405937760450944233503296, 5.69553977292108452936618179775

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