Properties

Label 8-546e4-1.1-c1e4-0-1
Degree $8$
Conductor $88873149456$
Sign $1$
Analytic cond. $361.309$
Root an. cond. $2.08802$
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  − 2·3-s − 2·4-s + 4·5-s + 6·7-s + 2·9-s + 4·12-s − 8·15-s + 3·16-s − 8·17-s − 8·20-s − 12·21-s − 6·27-s − 12·28-s + 24·35-s − 4·36-s − 28·37-s − 12·41-s − 24·43-s + 8·45-s − 8·47-s − 6·48-s + 18·49-s + 16·51-s + 16·60-s + 12·63-s − 4·64-s − 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s − 2.06·15-s + 3/4·16-s − 1.94·17-s − 1.78·20-s − 2.61·21-s − 1.15·27-s − 2.26·28-s + 4.05·35-s − 2/3·36-s − 4.60·37-s − 1.87·41-s − 3.65·43-s + 1.19·45-s − 1.16·47-s − 0.866·48-s + 18/7·49-s + 2.24·51-s + 2.06·60-s + 1.51·63-s − 1/2·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(361.309\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2067601150\)
\(L(\frac12)\) \(\approx\) \(0.2067601150\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$D_{4}$ \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 44 T^{2} + 2566 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
89$D_{4}$ \( ( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.73992541799218099319462437319, −7.57195611645069544541326832765, −7.38004250907892275496881373515, −6.89713450691884765911844281708, −6.78850499279001052125873378114, −6.50943660639169949410297023996, −6.32215619418720470922388997765, −6.04893702698268726148281776814, −5.88909163273873928532929461201, −5.21309367697032257563326574976, −5.20017607424063446404351680353, −5.07030838922238432529418584516, −5.00606979148762038657493776187, −4.97975177492659186506487025987, −4.24853661178930998788238765393, −4.23861201714527554878012662128, −3.60470593542013135407300783335, −3.40633856267428063218182071426, −3.33960899320937719408067851277, −2.32638493614287781146696039331, −2.04055636037187924835890136624, −1.91427100080912064527357997453, −1.54522890431399930719550629383, −1.46941403167887486019753201444, −0.14492481619184129478389763290, 0.14492481619184129478389763290, 1.46941403167887486019753201444, 1.54522890431399930719550629383, 1.91427100080912064527357997453, 2.04055636037187924835890136624, 2.32638493614287781146696039331, 3.33960899320937719408067851277, 3.40633856267428063218182071426, 3.60470593542013135407300783335, 4.23861201714527554878012662128, 4.24853661178930998788238765393, 4.97975177492659186506487025987, 5.00606979148762038657493776187, 5.07030838922238432529418584516, 5.20017607424063446404351680353, 5.21309367697032257563326574976, 5.88909163273873928532929461201, 6.04893702698268726148281776814, 6.32215619418720470922388997765, 6.50943660639169949410297023996, 6.78850499279001052125873378114, 6.89713450691884765911844281708, 7.38004250907892275496881373515, 7.57195611645069544541326832765, 7.73992541799218099319462437319

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