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Properties

Label 8-3e4-1.1-c18e4-0-0
Degree $8$
Conductor $81$
Sign $1$
Analytic cond. $1441.35$
Root an. cond. $2.48225$
Motivic weight $18$
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  + 1.58e4·3-s − 2.48e3·4-s − 9.57e7·7-s − 3.16e8·9-s − 3.93e7·12-s − 5.42e9·13-s − 1.87e10·16-s + 1.91e11·19-s − 1.52e12·21-s + 1.33e13·25-s − 7.90e12·27-s + 2.37e11·28-s + 3.57e13·31-s + 7.85e11·36-s − 4.75e14·37-s − 8.61e13·39-s + 1.59e15·43-s − 2.97e14·48-s − 3.85e14·49-s + 1.34e13·52-s + 3.03e15·57-s − 2.20e15·61-s + 3.03e16·63-s − 7.74e13·64-s − 4.13e16·67-s − 4.53e16·73-s + 2.11e17·75-s + ⋯
L(s)  = 1  + 0.806·3-s − 0.00946·4-s − 2.37·7-s − 0.817·9-s − 0.00763·12-s − 0.511·13-s − 0.272·16-s + 0.593·19-s − 1.91·21-s + 3.49·25-s − 1.03·27-s + 0.0224·28-s + 1.35·31-s + 0.00773·36-s − 3.65·37-s − 0.412·39-s + 3.18·43-s − 0.219·48-s − 0.236·49-s + 0.00484·52-s + 0.478·57-s − 0.188·61-s + 1.93·63-s − 0.00430·64-s − 1.51·67-s − 0.770·73-s + 2.81·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(1441.35\)
Root analytic conductor: \(2.48225\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 81,\ (\ :9, 9, 9, 9),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.5521639152\)
\(L(\frac12)\) \(\approx\) \(0.5521639152\)
\(L(10)\) 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$D_{4}$ \( 1 - 196 p^{4} T + 86678 p^{8} T^{2} - 196 p^{22} T^{3} + p^{36} T^{4} \)
good2$C_2^2 \wr C_2$ \( 1 + 155 p^{4} T^{2} + 1143897 p^{14} T^{4} + 155 p^{40} T^{6} + p^{72} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 2666638851668 p T^{2} + \)\(23\!\cdots\!82\)\( p^{5} T^{4} - 2666638851668 p^{37} T^{6} + p^{72} T^{8} \)
7$D_{4}$ \( ( 1 + 6838868 p T + 10583755945386 p^{3} T^{2} + 6838868 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 8040762618362030884 T^{2} + \)\(59\!\cdots\!66\)\( p^{2} T^{4} - 8040762618362030884 p^{36} T^{6} + p^{72} T^{8} \)
13$D_{4}$ \( ( 1 + 208700828 p T + 285201747815178822 p^{2} T^{2} + 208700828 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!20\)\( p^{2} T^{2} + \)\(71\!\cdots\!58\)\( p^{4} T^{4} - \)\(12\!\cdots\!20\)\( p^{38} T^{6} + p^{72} T^{8} \)
19$D_{4}$ \( ( 1 - 95708324740 T + \)\(12\!\cdots\!18\)\( T^{2} - 95708324740 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - \)\(89\!\cdots\!60\)\( T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(89\!\cdots\!60\)\( p^{36} T^{6} + p^{72} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(53\!\cdots\!84\)\( T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(53\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
31$D_{4}$ \( ( 1 - 17864207542804 T + \)\(10\!\cdots\!86\)\( T^{2} - 17864207542804 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 237649916751116 T + \)\(47\!\cdots\!98\)\( T^{2} + 237649916751116 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(21\!\cdots\!44\)\( T^{2} + \)\(34\!\cdots\!66\)\( T^{4} - \)\(21\!\cdots\!44\)\( p^{36} T^{6} + p^{72} T^{8} \)
43$D_{4}$ \( ( 1 - 799803662287396 T + \)\(47\!\cdots\!98\)\( T^{2} - 799803662287396 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(33\!\cdots\!40\)\( T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - \)\(33\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(13\!\cdots\!40\)\( T^{2} + \)\(66\!\cdots\!18\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(28\!\cdots\!44\)\( T^{2} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(28\!\cdots\!44\)\( p^{36} T^{6} + p^{72} T^{8} \)
61$D_{4}$ \( ( 1 + 1103634372941996 T + \)\(15\!\cdots\!66\)\( T^{2} + 1103634372941996 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 20665768039464956 T + \)\(15\!\cdots\!98\)\( T^{2} + 20665768039464956 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(35\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(35\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
73$D_{4}$ \( ( 1 + 22694977381228124 T + \)\(62\!\cdots\!78\)\( T^{2} + 22694977381228124 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 1122565721708180 p T + \)\(29\!\cdots\!58\)\( T^{2} + 1122565721708180 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!40\)\( T^{2} + \)\(64\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(22\!\cdots\!84\)\( T^{2} + \)\(25\!\cdots\!86\)\( T^{4} - \)\(22\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
97$D_{4}$ \( ( 1 + 1745782267257380156 T + \)\(18\!\cdots\!98\)\( T^{2} + 1745782267257380156 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
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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

−15.79915870126307033203670850997, −15.55867620595009917719742416165, −14.66553677059359858556372116360, −14.35510480299154864627661117601, −13.91955782710433412103737814269, −13.31277110908984203301800330263, −12.80916506798079875930602272471, −12.33252076187058749702464154719, −12.01801297980422112153158910080, −10.75062418148835970374885015041, −10.69116032561887396356713062181, −9.549559944850293134810634436874, −9.539596258056412087260220552730, −8.654058992399360153589039282056, −8.459791233845743354907479843072, −7.04734512498114625163189500203, −6.97578087424956301278800764643, −6.15854778481231968025290218798, −5.36882103519765579444020504273, −4.47006469852609182577849627871, −3.18651791608639209177268081610, −3.10475372478871522884758767004, −2.64563251124459065798211352237, −1.24154417391559903032098438629, −0.21279680117724765709893767246, 0.21279680117724765709893767246, 1.24154417391559903032098438629, 2.64563251124459065798211352237, 3.10475372478871522884758767004, 3.18651791608639209177268081610, 4.47006469852609182577849627871, 5.36882103519765579444020504273, 6.15854778481231968025290218798, 6.97578087424956301278800764643, 7.04734512498114625163189500203, 8.459791233845743354907479843072, 8.654058992399360153589039282056, 9.539596258056412087260220552730, 9.549559944850293134810634436874, 10.69116032561887396356713062181, 10.75062418148835970374885015041, 12.01801297980422112153158910080, 12.33252076187058749702464154719, 12.80916506798079875930602272471, 13.31277110908984203301800330263, 13.91955782710433412103737814269, 14.35510480299154864627661117601, 14.66553677059359858556372116360, 15.55867620595009917719742416165, 15.79915870126307033203670850997

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