Properties

Label 8-6e12-1.1-c1e4-0-1
Degree $8$
Conductor $2176782336$
Sign $1$
Analytic cond. $8.84959$
Root an. cond. $1.31330$
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  − 4·4-s − 4·7-s + 12·16-s − 2·25-s + 16·28-s + 20·31-s + 18·49-s − 32·64-s − 28·73-s − 40·79-s − 4·97-s + 8·100-s + 56·103-s − 48·112-s + 10·121-s − 80·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + ⋯
L(s)  = 1  − 2·4-s − 1.51·7-s + 3·16-s − 2/5·25-s + 3.02·28-s + 3.59·31-s + 18/7·49-s − 4·64-s − 3.27·73-s − 4.50·79-s − 0.406·97-s + 4/5·100-s + 5.51·103-s − 4.53·112-s + 0.909·121-s − 7.18·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(8.84959\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6144994865\)
\(L(\frac12)\) \(\approx\) \(0.6144994865\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
good5$C_2^3$ \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^3$ \( 1 - 94 T^{2} + 6027 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.966689318783669584270217950417, −8.890519749902944527139694254697, −8.406152479796825356247832892286, −8.353079900643625565664156056767, −8.293053947003363519032759692182, −7.54840850903100823415266873109, −7.43320751867388141567271288795, −7.29094383179781281497956501507, −6.88077449914203091387254420731, −6.45991083049453539210012708206, −6.08030678483194142331386839723, −5.89356311258118591278667320640, −5.87698713383913919953825076736, −5.43491966857946917100599255991, −4.93396269287804346789627372966, −4.56654284062922243129795878754, −4.35202359523443786773244670771, −4.32242328711747486569045006635, −3.80586137743114081422903519392, −3.23425853026846556284851884959, −3.02891429392479997191239342200, −2.92247024654684937327965633141, −2.09347155128834804444612063635, −1.20357386349153630305809421009, −0.55514023329223631181035262769, 0.55514023329223631181035262769, 1.20357386349153630305809421009, 2.09347155128834804444612063635, 2.92247024654684937327965633141, 3.02891429392479997191239342200, 3.23425853026846556284851884959, 3.80586137743114081422903519392, 4.32242328711747486569045006635, 4.35202359523443786773244670771, 4.56654284062922243129795878754, 4.93396269287804346789627372966, 5.43491966857946917100599255991, 5.87698713383913919953825076736, 5.89356311258118591278667320640, 6.08030678483194142331386839723, 6.45991083049453539210012708206, 6.88077449914203091387254420731, 7.29094383179781281497956501507, 7.43320751867388141567271288795, 7.54840850903100823415266873109, 8.293053947003363519032759692182, 8.353079900643625565664156056767, 8.406152479796825356247832892286, 8.890519749902944527139694254697, 8.966689318783669584270217950417

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