Properties

Label 8-260e4-1.1-c1e4-0-7
Degree $8$
Conductor $4569760000$
Sign $1$
Analytic cond. $18.5781$
Root an. cond. $1.44087$
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  + 2·2-s + 2·4-s + 4·5-s + 4·8-s + 8·10-s + 6·13-s + 8·16-s − 12·17-s + 8·20-s + 5·25-s + 12·26-s + 30·29-s + 8·32-s − 24·34-s − 22·37-s + 16·40-s − 8·41-s + 10·50-s + 12·52-s − 18·53-s + 60·58-s + 12·61-s + 8·64-s + 24·65-s − 24·68-s + 22·73-s − 44·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s + 1.41·8-s + 2.52·10-s + 1.66·13-s + 2·16-s − 2.91·17-s + 1.78·20-s + 25-s + 2.35·26-s + 5.57·29-s + 1.41·32-s − 4.11·34-s − 3.61·37-s + 2.52·40-s − 1.24·41-s + 1.41·50-s + 1.66·52-s − 2.47·53-s + 7.87·58-s + 1.53·61-s + 64-s + 2.97·65-s − 2.91·68-s + 2.57·73-s − 5.11·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(18.5781\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.504299306\)
\(L(\frac12)\) \(\approx\) \(6.504299306\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
7$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
43$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
67$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
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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

−8.773730036935169036906822027648, −8.338545543240862976115667870552, −8.182670313163934404222519672721, −8.112754915106078788657934979963, −7.980125766356866214935445805889, −7.02685138958895284170952785180, −6.77625515828027455135238912794, −6.76196100470773627946624793768, −6.55723470307230794848023005047, −6.44268813216608547240542311524, −6.24057580277697110549768917542, −5.59502130895678779070753013944, −5.29982756770589490527677548566, −5.26798049548612350813225944030, −4.93234467787710136666877367044, −4.59839954173319992106380533470, −4.26406273546306389509324680586, −4.14966441261359599347151776550, −3.51391548556867953711160136834, −3.35362846456650004915733170313, −2.72329521910870590961641034107, −2.56503088112471040935254856178, −1.95100325114145574096345593038, −1.65670186023270692898230900837, −1.18967089992317893592939597282, 1.18967089992317893592939597282, 1.65670186023270692898230900837, 1.95100325114145574096345593038, 2.56503088112471040935254856178, 2.72329521910870590961641034107, 3.35362846456650004915733170313, 3.51391548556867953711160136834, 4.14966441261359599347151776550, 4.26406273546306389509324680586, 4.59839954173319992106380533470, 4.93234467787710136666877367044, 5.26798049548612350813225944030, 5.29982756770589490527677548566, 5.59502130895678779070753013944, 6.24057580277697110549768917542, 6.44268813216608547240542311524, 6.55723470307230794848023005047, 6.76196100470773627946624793768, 6.77625515828027455135238912794, 7.02685138958895284170952785180, 7.980125766356866214935445805889, 8.112754915106078788657934979963, 8.182670313163934404222519672721, 8.338545543240862976115667870552, 8.773730036935169036906822027648

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