Properties

Label 8-288e4-1.1-c1e4-0-5
Degree $8$
Conductor $6879707136$
Sign $1$
Analytic cond. $27.9690$
Root an. cond. $1.51647$
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 + 3·9-s + 18·11-s + 4·19-s + 10·25-s − 10·27-s − 36·33-s + 18·41-s + 10·43-s − 14·49-s − 8·57-s − 18·59-s − 14·67-s − 4·73-s − 20·75-s + 20·81-s − 10·97-s + 54·99-s + 169·121-s − 36·123-s + 127-s − 20·129-s + 131-s + 137-s + 139-s + 28·147-s + 149-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 5.42·11-s + 0.917·19-s + 2·25-s − 1.92·27-s − 6.26·33-s + 2.81·41-s + 1.52·43-s − 2·49-s − 1.05·57-s − 2.34·59-s − 1.71·67-s − 0.468·73-s − 2.30·75-s + 20/9·81-s − 1.01·97-s + 5.42·99-s + 15.3·121-s − 3.24·123-s + 0.0887·127-s − 1.76·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·147-s + 0.0819·149-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(27.9690\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.236607894\)
\(L(\frac12)\) \(\approx\) \(2.236607894\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
13$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
43$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
61$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 14 T + p T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
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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

−8.827924567889045885301099642545, −8.501300124998138230446948964577, −7.920531447084458532137297802157, −7.76863507082012229687281988011, −7.55329826700543950434768949724, −7.10091141452713149675408339067, −6.83099914402236818248649814665, −6.78457114150948124790182012393, −6.63982992344918068078870872196, −6.01120858957617329698265157798, −6.00223255077593968430739697067, −5.97829348138928179141918463939, −5.64910693417385723551842872037, −4.79838545457740215565684306812, −4.79068267244108707273614832376, −4.48742206388738385160313811509, −4.18584635841734405353329323843, −3.87570584615837684313082337805, −3.68849337934032410300149581862, −3.26201932961832851480031272467, −2.90739748907543855475989015453, −2.14798322928933584975022696282, −1.44872290482539340360860925926, −1.24575039547965681227910336117, −1.06162082961248363009660896400, 1.06162082961248363009660896400, 1.24575039547965681227910336117, 1.44872290482539340360860925926, 2.14798322928933584975022696282, 2.90739748907543855475989015453, 3.26201932961832851480031272467, 3.68849337934032410300149581862, 3.87570584615837684313082337805, 4.18584635841734405353329323843, 4.48742206388738385160313811509, 4.79068267244108707273614832376, 4.79838545457740215565684306812, 5.64910693417385723551842872037, 5.97829348138928179141918463939, 6.00223255077593968430739697067, 6.01120858957617329698265157798, 6.63982992344918068078870872196, 6.78457114150948124790182012393, 6.83099914402236818248649814665, 7.10091141452713149675408339067, 7.55329826700543950434768949724, 7.76863507082012229687281988011, 7.920531447084458532137297802157, 8.501300124998138230446948964577, 8.827924567889045885301099642545

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