Properties

Label 8-4005e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.573\times 10^{14}$
Sign $1$
Analytic cond. $1.04596\times 10^{6}$
Root an. cond. $5.65509$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 4·5-s − 7·7-s + 8-s − 5·13-s + 16-s + 13·17-s − 10·19-s − 12·20-s − 3·23-s + 10·25-s + 21·28-s − 2·29-s − 2·31-s − 4·32-s − 28·35-s − 9·37-s + 4·40-s − 2·41-s − 19·43-s − 5·47-s + 7·49-s + 15·52-s + 3·53-s − 7·56-s − 2·59-s − 4·61-s + ⋯
L(s)  = 1  − 3/2·4-s + 1.78·5-s − 2.64·7-s + 0.353·8-s − 1.38·13-s + 1/4·16-s + 3.15·17-s − 2.29·19-s − 2.68·20-s − 0.625·23-s + 2·25-s + 3.96·28-s − 0.371·29-s − 0.359·31-s − 0.707·32-s − 4.73·35-s − 1.47·37-s + 0.632·40-s − 0.312·41-s − 2.89·43-s − 0.729·47-s + 49-s + 2.08·52-s + 0.412·53-s − 0.935·56-s − 0.260·59-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 89^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 89^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 89^{4}\)
Sign: $1$
Analytic conductor: \(1.04596\times 10^{6}\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 89^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
89$C_1$ \( ( 1 + T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 3 T^{2} - T^{3} + p^{3} T^{4} - p T^{5} + 3 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + p T + 6 p T^{2} + 152 T^{3} + 486 T^{4} + 152 p T^{5} + 6 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 24 T^{2} + 8 T^{3} + 350 T^{4} + 8 p T^{5} + 24 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 5 T + 42 T^{2} + 112 T^{3} + 648 T^{4} + 112 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 13 T + 122 T^{2} - 750 T^{3} + 3614 T^{4} - 750 p T^{5} + 122 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 10 T + 96 T^{2} + 554 T^{3} + 2910 T^{4} + 554 p T^{5} + 96 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 3 T + 50 T^{2} + 79 T^{3} + 1154 T^{4} + 79 p T^{5} + 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 2 T + 79 T^{2} + 85 T^{3} + 2941 T^{4} + 85 p T^{5} + 79 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 2 T + 92 T^{2} + 154 T^{3} + 3910 T^{4} + 154 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 9 T + 128 T^{2} + 742 T^{3} + 6348 T^{4} + 742 p T^{5} + 128 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 2 T + 161 T^{2} + 241 T^{3} + 9841 T^{4} + 241 p T^{5} + 161 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 19 T + 212 T^{2} + 1732 T^{3} + 12390 T^{4} + 1732 p T^{5} + 212 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 5 T + 144 T^{2} + 510 T^{3} + 190 p T^{4} + 510 p T^{5} + 144 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 3 T + 52 T^{2} + 1036 T^{4} + 52 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 2 T + 179 T^{2} + 269 T^{3} + 14137 T^{4} + 269 p T^{5} + 179 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 4 T + 104 T^{2} + 740 T^{3} + 8254 T^{4} + 740 p T^{5} + 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 15 T + 316 T^{2} + 3031 T^{3} + 33302 T^{4} + 3031 p T^{5} + 316 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 6 T + 260 T^{2} - 1094 T^{3} + 26582 T^{4} - 1094 p T^{5} + 260 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 21 T + 392 T^{2} + 4759 T^{3} + 46638 T^{4} + 4759 p T^{5} + 392 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 20 T + 375 T^{2} + 4591 T^{3} + 46835 T^{4} + 4591 p T^{5} + 375 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 9 T + 222 T^{2} - 2097 T^{3} + 24090 T^{4} - 2097 p T^{5} + 222 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 17 T + 362 T^{2} + 4283 T^{3} + 51834 T^{4} + 4283 p T^{5} + 362 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.34434548506716005334630900462, −6.30524051058099236053950284053, −5.93480651203763633901842528590, −5.69333514950574571630780356292, −5.66425796202354289310518930399, −5.34289337066975392829892033225, −5.17049670234724065481686562437, −5.00007536439058911221619875776, −4.97898603724571113075723656348, −4.50395435522721923500352104959, −4.40823052799528889413178143506, −4.16356660433129033711492239075, −4.10825691479581875255526505808, −3.48736523451728055000054106477, −3.38036900921596979489712197797, −3.35405572993031034727139376525, −3.26064616818322372542165152369, −2.84797924096663676487797722636, −2.77124037225154948895341335245, −2.20960909003248755388655957192, −2.13598424001074892452491894956, −1.96999653325533037337038288645, −1.39821328060832024916949273209, −1.32552471067814806108772766722, −1.17123567357001020992146879395, 0, 0, 0, 0, 1.17123567357001020992146879395, 1.32552471067814806108772766722, 1.39821328060832024916949273209, 1.96999653325533037337038288645, 2.13598424001074892452491894956, 2.20960909003248755388655957192, 2.77124037225154948895341335245, 2.84797924096663676487797722636, 3.26064616818322372542165152369, 3.35405572993031034727139376525, 3.38036900921596979489712197797, 3.48736523451728055000054106477, 4.10825691479581875255526505808, 4.16356660433129033711492239075, 4.40823052799528889413178143506, 4.50395435522721923500352104959, 4.97898603724571113075723656348, 5.00007536439058911221619875776, 5.17049670234724065481686562437, 5.34289337066975392829892033225, 5.66425796202354289310518930399, 5.69333514950574571630780356292, 5.93480651203763633901842528590, 6.30524051058099236053950284053, 6.34434548506716005334630900462

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