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Properties

Label 8-5472e4-1.1-c1e4-0-7
Degree $8$
Conductor $8.966\times 10^{14}$
Sign $1$
Analytic cond. $3.64496\times 10^{6}$
Root an. cond. $6.61015$
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  − 5-s − 7-s − 7·11-s + 10·13-s − 5·17-s + 4·19-s + 8·23-s − 25-s − 10·29-s − 6·31-s + 35-s + 14·37-s + 2·41-s + 3·43-s + 3·47-s − 10·49-s − 4·53-s + 7·55-s − 20·59-s − 3·61-s − 10·65-s + 8·67-s + 30·71-s + 9·73-s + 7·77-s + 10·79-s − 4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 2.11·11-s + 2.77·13-s − 1.21·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.85·29-s − 1.07·31-s + 0.169·35-s + 2.30·37-s + 0.312·41-s + 0.457·43-s + 0.437·47-s − 1.42·49-s − 0.549·53-s + 0.943·55-s − 2.60·59-s − 0.384·61-s − 1.24·65-s + 0.977·67-s + 3.56·71-s + 1.05·73-s + 0.797·77-s + 1.12·79-s − 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(3.64496\times 10^{6}\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.344412302\)
\(L(\frac12)\) \(\approx\) \(5.344412302\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + T + 2 T^{2} - 9 T^{3} + 2 T^{4} - 9 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + T + 11 T^{2} + 40 T^{3} + 60 T^{4} + 40 p T^{5} + 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 4 p T^{2} + 203 T^{3} + 742 T^{4} + 203 p T^{5} + 4 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 59 T^{2} - 232 T^{3} + 852 T^{4} - 232 p T^{5} + 59 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + p T^{2} + 114 T^{3} + 698 T^{4} + 114 p T^{5} + p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 75 T^{2} - 420 T^{3} + 2456 T^{4} - 420 p T^{5} + 75 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 123 T^{2} + 712 T^{3} + 5108 T^{4} + 712 p T^{5} + 123 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 68 T^{2} + 142 T^{3} + 1782 T^{4} + 142 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 152 T^{2} - 1018 T^{3} + 7006 T^{4} - 1018 p T^{5} + 152 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 92 T^{2} - 54 T^{3} + 4694 T^{4} - 54 p T^{5} + 92 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 140 T^{2} - 291 T^{3} + 8278 T^{4} - 291 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 80 T^{2} - 247 T^{3} + 3038 T^{4} - 247 p T^{5} + 80 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 143 T^{2} + 252 T^{3} + 9032 T^{4} + 252 p T^{5} + 143 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 321 T^{2} + 3390 T^{3} + 30764 T^{4} + 3390 p T^{5} + 321 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 178 T^{2} + 521 T^{3} + 14618 T^{4} + 521 p T^{5} + 178 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 193 T^{2} - 1006 T^{3} + 16060 T^{4} - 1006 p T^{5} + 193 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 552 T^{2} - 6814 T^{3} + 65870 T^{4} - 6814 p T^{5} + 552 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 245 T^{2} - 1662 T^{3} + 25114 T^{4} - 1662 p T^{5} + 245 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 216 T^{2} - 1082 T^{3} + 18510 T^{4} - 1082 p T^{5} + 216 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 216 T^{2} + 212 T^{3} + 21054 T^{4} + 212 p T^{5} + 216 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 152 T^{2} - 224 T^{3} - 4082 T^{4} - 224 p T^{5} + 152 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 328 T^{2} + 1658 T^{3} + 45422 T^{4} + 1658 p T^{5} + 328 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.81325015412971544816459589352, −5.67756115309310424932057230418, −5.42862153080607770822644292627, −5.07158529252176442057555026014, −4.90946288793069050499894338940, −4.84236942476146888101873164217, −4.64775118422532926659731604040, −4.49030519823577381513309839952, −4.24316500237545842419887717422, −3.76189749426023620881916728087, −3.67582250694019429363216354986, −3.55881674363583535585521589590, −3.51919870085167850674973397722, −3.24177366327888211523928082458, −2.94364180151707286965636413051, −2.71565441017914983101763142581, −2.63970398911758114660672456437, −2.13684230202597307160489410611, −1.92783458749699349504749951061, −1.89925409662574441906238265598, −1.60364571118123648364589042244, −1.03941637650464433690852259419, −0.827462776982066222119245862668, −0.48992422295841945154161943574, −0.47825230995625453088936185001, 0.47825230995625453088936185001, 0.48992422295841945154161943574, 0.827462776982066222119245862668, 1.03941637650464433690852259419, 1.60364571118123648364589042244, 1.89925409662574441906238265598, 1.92783458749699349504749951061, 2.13684230202597307160489410611, 2.63970398911758114660672456437, 2.71565441017914983101763142581, 2.94364180151707286965636413051, 3.24177366327888211523928082458, 3.51919870085167850674973397722, 3.55881674363583535585521589590, 3.67582250694019429363216354986, 3.76189749426023620881916728087, 4.24316500237545842419887717422, 4.49030519823577381513309839952, 4.64775118422532926659731604040, 4.84236942476146888101873164217, 4.90946288793069050499894338940, 5.07158529252176442057555026014, 5.42862153080607770822644292627, 5.67756115309310424932057230418, 5.81325015412971544816459589352

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