Properties

Label 6-507e3-1.1-c1e3-0-3
Degree $6$
Conductor $130323843$
Sign $-1$
Analytic cond. $66.3521$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s − 3·4-s − 4·5-s − 3·6-s − 10·7-s + 4·8-s + 6·9-s + 4·10-s + 11-s − 9·12-s + 10·14-s − 12·15-s + 3·16-s − 7·17-s − 6·18-s − 11·19-s + 12·20-s − 30·21-s − 22-s + 2·23-s + 12·24-s − 2·25-s + 10·27-s + 30·28-s − 8·29-s + 12·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 3/2·4-s − 1.78·5-s − 1.22·6-s − 3.77·7-s + 1.41·8-s + 2·9-s + 1.26·10-s + 0.301·11-s − 2.59·12-s + 2.67·14-s − 3.09·15-s + 3/4·16-s − 1.69·17-s − 1.41·18-s − 2.52·19-s + 2.68·20-s − 6.54·21-s − 0.213·22-s + 0.417·23-s + 2.44·24-s − 2/5·25-s + 1.92·27-s + 5.66·28-s − 1.48·29-s + 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(66.3521\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{3} \)
13 \( 1 \)
good2$A_4\times C_2$ \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 10 T + 52 T^{2} + 169 T^{3} + 52 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - T + 3 T^{2} + 21 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 11 T + 67 T^{2} + 305 T^{3} + 67 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 2 T + 26 T^{2} - 175 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 8 T + 70 T^{2} + 299 T^{3} + 70 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 14 T + 174 T^{2} + 1127 T^{3} + 174 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + T + 121 T^{2} + 81 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 3 T + 104 T^{2} + 287 T^{3} + 104 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 9 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 1596 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 5 T + 179 T^{2} + 573 T^{3} + 179 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 6 T + 134 T^{2} - 391 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 18 T + 320 T^{2} + 2795 T^{3} + 320 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 16 T + 304 T^{2} - 2613 T^{3} + 304 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 5 T + 259 T^{2} - 891 T^{3} + 259 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 5 T + 122 T^{2} - 219 T^{3} + 122 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.04622159425429106782801160423, −9.545318827431046671283032706021, −9.387936655007998413882719602178, −9.328664358982393923814901433342, −8.901916956417484952978620653598, −8.827944397827818722666709817824, −8.688382530171444118859210317137, −8.134379072855960489066306656933, −7.925097567000946502220566869297, −7.62837194704805766727118585463, −7.05861904768499673437375562679, −6.89865178120035937609479784441, −6.73304836595896926602913362769, −6.42142572689273810047465156245, −5.75526842125667722901185388159, −5.66175844200956706116341339631, −4.67470831562182051652699786418, −4.32191340047312199842370424717, −4.26503047454121388669926092409, −3.69110187342332593466895818269, −3.52848953311962123409972105103, −3.52723829448976754049092031553, −2.91060722505024791174012062239, −2.24268984618268840451525788852, −1.85376290648341535326764824068, 0, 0, 0, 1.85376290648341535326764824068, 2.24268984618268840451525788852, 2.91060722505024791174012062239, 3.52723829448976754049092031553, 3.52848953311962123409972105103, 3.69110187342332593466895818269, 4.26503047454121388669926092409, 4.32191340047312199842370424717, 4.67470831562182051652699786418, 5.66175844200956706116341339631, 5.75526842125667722901185388159, 6.42142572689273810047465156245, 6.73304836595896926602913362769, 6.89865178120035937609479784441, 7.05861904768499673437375562679, 7.62837194704805766727118585463, 7.925097567000946502220566869297, 8.134379072855960489066306656933, 8.688382530171444118859210317137, 8.827944397827818722666709817824, 8.901916956417484952978620653598, 9.328664358982393923814901433342, 9.387936655007998413882719602178, 9.545318827431046671283032706021, 10.04622159425429106782801160423

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