Properties

Label 8-6e12-1.1-c2e4-0-0
Degree $8$
Conductor $2176782336$
Sign $1$
Analytic cond. $1199.92$
Root an. cond. $2.42602$
Motivic weight $2$
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  − 12·7-s − 8·13-s + 32·19-s + 66·25-s − 4·31-s − 144·37-s − 56·43-s + 38·49-s + 208·61-s − 232·67-s − 284·73-s − 296·79-s + 96·91-s + 44·97-s + 136·103-s − 272·109-s + 258·121-s + 127-s + 131-s − 384·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.71·7-s − 0.615·13-s + 1.68·19-s + 2.63·25-s − 0.129·31-s − 3.89·37-s − 1.30·43-s + 0.775·49-s + 3.40·61-s − 3.46·67-s − 3.89·73-s − 3.74·79-s + 1.05·91-s + 0.453·97-s + 1.32·103-s − 2.49·109-s + 2.13·121-s + 0.00787·127-s + 0.00763·131-s − 2.88·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(1199.92\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1521982888\)
\(L(\frac12)\) \(\approx\) \(0.1521982888\)
\(L(2)\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 - 66 T^{2} + 2051 T^{4} - 66 p^{4} T^{6} + p^{8} T^{8} \)
7$D_{4}$ \( ( 1 + 6 T + 5 p T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 258 T^{2} + 35555 T^{4} - 258 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 828 T^{2} + 320006 T^{4} - 828 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 - 16 T + 498 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 228 T^{2} + 111878 T^{4} - 228 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2}( 1 + 40 T + p^{2} T^{2} )^{2} \)
31$D_{4}$ \( ( 1 + 2 T + 1851 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 72 T + 3746 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 1252 T^{2} + 4550406 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 28 T + 3606 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3004 T^{2} + 3886854 T^{4} - 3004 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 1410 T^{2} - 7776061 T^{4} - 1410 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 732 T^{2} + 16240166 T^{4} - 732 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 104 T + 5538 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 116 T + 12054 T^{2} + 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 17916 T^{2} + 130032326 T^{4} - 17916 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 142 T + 13107 T^{2} + 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 148 T + 16806 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 14898 T^{2} + 120543251 T^{4} - 14898 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 14908 T^{2} + 171715398 T^{4} - 14908 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 - 22 T + 14331 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−8.854831710587900713715848969275, −8.759745376845820830000951615831, −8.304520835761411012804160708155, −8.051348791001884862673035726873, −7.47795705837422618906243594644, −7.19670981509660302210587104820, −7.14755742874091157850481962171, −7.00009272728732160845430687009, −6.64025977040393390302795049219, −6.50087109378114592712649509545, −5.83879449562325201496636663283, −5.74070309820807726741042242165, −5.47487800578571047940636358809, −5.14073556379621651848391901673, −4.69668050757654793678750660705, −4.63827959934901184803169099348, −4.08912035238932704261451643263, −3.49868970438358132230939231147, −3.23978817656453217055289039956, −3.22031679948691767909826366995, −2.78713938538179779123173088045, −2.33912834091557295278612245573, −1.36857783633488885672175505706, −1.36499465014207392076189423723, −0.11371717290972891094726224560, 0.11371717290972891094726224560, 1.36499465014207392076189423723, 1.36857783633488885672175505706, 2.33912834091557295278612245573, 2.78713938538179779123173088045, 3.22031679948691767909826366995, 3.23978817656453217055289039956, 3.49868970438358132230939231147, 4.08912035238932704261451643263, 4.63827959934901184803169099348, 4.69668050757654793678750660705, 5.14073556379621651848391901673, 5.47487800578571047940636358809, 5.74070309820807726741042242165, 5.83879449562325201496636663283, 6.50087109378114592712649509545, 6.64025977040393390302795049219, 7.00009272728732160845430687009, 7.14755742874091157850481962171, 7.19670981509660302210587104820, 7.47795705837422618906243594644, 8.051348791001884862673035726873, 8.304520835761411012804160708155, 8.759745376845820830000951615831, 8.854831710587900713715848969275

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