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Properties

Label 8-3e8-1.1-c19e4-0-0
Degree $8$
Conductor $6561$
Sign $1$
Analytic cond. $179854.$
Root an. cond. $4.53800$
Motivic weight $19$
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  − 6.50e5·4-s + 1.66e8·7-s + 5.31e10·13-s + 1.82e11·16-s + 3.76e12·19-s + 3.97e12·25-s − 1.08e14·28-s + 4.71e14·31-s − 2.22e15·37-s − 1.22e16·43-s + 7.57e15·49-s − 3.45e16·52-s + 1.60e17·61-s − 1.40e17·64-s − 4.86e17·67-s − 1.30e17·73-s − 2.45e18·76-s + 3.24e18·79-s + 8.83e18·91-s − 4.17e17·97-s − 2.58e18·100-s + 1.36e19·103-s + 6.21e19·109-s + 3.02e19·112-s + 3.04e19·121-s − 3.06e20·124-s + 127-s + ⋯
L(s)  = 1  − 1.24·4-s + 1.55·7-s + 1.39·13-s + 0.662·16-s + 2.67·19-s + 0.208·25-s − 1.93·28-s + 3.20·31-s − 2.81·37-s − 3.72·43-s + 0.664·49-s − 1.72·52-s + 1.75·61-s − 0.976·64-s − 2.18·67-s − 0.259·73-s − 3.32·76-s + 3.04·79-s + 2.16·91-s − 0.0558·97-s − 0.258·100-s + 1.03·103-s + 2.74·109-s + 1.03·112-s + 0.497·121-s − 3.97·124-s + 4.17·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6561\)    =    \(3^{8}\)
Sign: $1$
Analytic conductor: \(179854.\)
Root analytic conductor: \(4.53800\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6561,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(2.808679178\)
\(L(\frac12)\) \(\approx\) \(2.808679178\)
\(L(\frac{21}{2})\) 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)$
bad3 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 + 81289 p^{3} T^{2} + 29384697 p^{13} T^{4} + 81289 p^{41} T^{6} + p^{76} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 795488126108 p T^{2} - \)\(33\!\cdots\!82\)\( p^{5} T^{4} - 795488126108 p^{39} T^{6} + p^{76} T^{8} \)
7$D_{4}$ \( ( 1 - 83136040 T + 134314140019614 p^{2} T^{2} - 83136040 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 2764417357493218076 p T^{2} + \)\(46\!\cdots\!06\)\( p^{3} T^{4} - 2764417357493218076 p^{39} T^{6} + p^{76} T^{8} \)
13$D_{4}$ \( ( 1 - 2044526020 p T + \)\(17\!\cdots\!58\)\( p T^{2} - 2044526020 p^{20} T^{3} + p^{38} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 + \)\(36\!\cdots\!56\)\( p T^{2} + \)\(41\!\cdots\!38\)\( p^{3} T^{4} + \)\(36\!\cdots\!56\)\( p^{39} T^{6} + p^{76} T^{8} \)
19$D_{4}$ \( ( 1 - 1884320709232 T + \)\(45\!\cdots\!14\)\( T^{2} - 1884320709232 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + \)\(15\!\cdots\!08\)\( T^{2} + \)\(17\!\cdots\!54\)\( T^{4} + \)\(15\!\cdots\!08\)\( p^{38} T^{6} + p^{76} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + \)\(17\!\cdots\!76\)\( T^{2} + \)\(14\!\cdots\!66\)\( T^{4} + \)\(17\!\cdots\!76\)\( p^{38} T^{6} + p^{76} T^{8} \)
31$D_{4}$ \( ( 1 - 235795954514536 T + \)\(57\!\cdots\!66\)\( T^{2} - 235795954514536 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 30115090384340 p T + \)\(10\!\cdots\!34\)\( p^{2} T^{2} + 30115090384340 p^{20} T^{3} + p^{38} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + \)\(11\!\cdots\!44\)\( T^{2} + \)\(64\!\cdots\!26\)\( T^{4} + \)\(11\!\cdots\!44\)\( p^{38} T^{6} + p^{76} T^{8} \)
43$D_{4}$ \( ( 1 + 6131163137396240 T + \)\(31\!\cdots\!14\)\( T^{2} + 6131163137396240 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + \)\(11\!\cdots\!92\)\( T^{2} - \)\(52\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!92\)\( p^{38} T^{6} + p^{76} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(41\!\cdots\!92\)\( T^{2} + \)\(46\!\cdots\!94\)\( T^{4} - \)\(41\!\cdots\!92\)\( p^{38} T^{6} + p^{76} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 + \)\(15\!\cdots\!56\)\( T^{2} + \)\(97\!\cdots\!26\)\( T^{4} + \)\(15\!\cdots\!56\)\( p^{38} T^{6} + p^{76} T^{8} \)
61$D_{4}$ \( ( 1 - 80023135652803564 T + \)\(14\!\cdots\!06\)\( T^{2} - 80023135652803564 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 243276932929153760 T + \)\(10\!\cdots\!06\)\( T^{2} + 243276932929153760 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 + \)\(15\!\cdots\!24\)\( T^{2} + \)\(31\!\cdots\!66\)\( T^{4} + \)\(15\!\cdots\!24\)\( p^{38} T^{6} + p^{76} T^{8} \)
73$D_{4}$ \( ( 1 + 65336433442154420 T + \)\(43\!\cdots\!74\)\( T^{2} + 65336433442154420 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 1623394896860607208 T + \)\(25\!\cdots\!54\)\( T^{2} - 1623394896860607208 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 + \)\(10\!\cdots\!28\)\( T^{2} + \)\(42\!\cdots\!14\)\( T^{4} + \)\(10\!\cdots\!28\)\( p^{38} T^{6} + p^{76} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + \)\(25\!\cdots\!36\)\( T^{2} + \)\(39\!\cdots\!86\)\( T^{4} + \)\(25\!\cdots\!36\)\( p^{38} T^{6} + p^{76} T^{8} \)
97$D_{4}$ \( ( 1 + 208943439542479460 T + \)\(10\!\cdots\!66\)\( T^{2} + 208943439542479460 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
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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

−11.38038777796012723621284018496, −11.22034935918262454085207816277, −10.26147885830418433721141465448, −10.25788555109282955452268159638, −9.854775753113634847058529266114, −9.245063620457830177777395277366, −8.627493045174101912330532778160, −8.523745649350380193167459289641, −8.312881631027681663251785709172, −7.60682674464355244031145488339, −7.38874197930855208577608591949, −6.43363357016295227552908590420, −6.41735804543968868147173022607, −5.38347937947970676428649871863, −5.21804184854692112281831658017, −4.74267610460094078109004980803, −4.65893562477493081315816142854, −3.74060162322891924699048432216, −3.33892894771027809057727199086, −3.16271386063107038118506767068, −2.18293808923156101188963703422, −1.57199960636773876024124659904, −1.16138182504981284509080165841, −1.01347970614729321211454102699, −0.29246756295489581294833216863, 0.29246756295489581294833216863, 1.01347970614729321211454102699, 1.16138182504981284509080165841, 1.57199960636773876024124659904, 2.18293808923156101188963703422, 3.16271386063107038118506767068, 3.33892894771027809057727199086, 3.74060162322891924699048432216, 4.65893562477493081315816142854, 4.74267610460094078109004980803, 5.21804184854692112281831658017, 5.38347937947970676428649871863, 6.41735804543968868147173022607, 6.43363357016295227552908590420, 7.38874197930855208577608591949, 7.60682674464355244031145488339, 8.312881631027681663251785709172, 8.523745649350380193167459289641, 8.627493045174101912330532778160, 9.245063620457830177777395277366, 9.854775753113634847058529266114, 10.25788555109282955452268159638, 10.26147885830418433721141465448, 11.22034935918262454085207816277, 11.38038777796012723621284018496

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