Properties

Label 6-8470e3-1.1-c1e3-0-3
Degree $6$
Conductor $607645423000$
Sign $1$
Analytic cond. $309372.$
Root an. cond. $8.22394$
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  + 3·2-s + 2·3-s + 6·4-s + 3·5-s + 6·6-s − 3·7-s + 10·8-s + 4·9-s + 9·10-s + 12·12-s − 4·13-s − 9·14-s + 6·15-s + 15·16-s − 17-s + 12·18-s + 3·19-s + 18·20-s − 6·21-s + 10·23-s + 20·24-s + 6·25-s − 12·26-s + 6·27-s − 18·28-s + 6·29-s + 18·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 3·4-s + 1.34·5-s + 2.44·6-s − 1.13·7-s + 3.53·8-s + 4/3·9-s + 2.84·10-s + 3.46·12-s − 1.10·13-s − 2.40·14-s + 1.54·15-s + 15/4·16-s − 0.242·17-s + 2.82·18-s + 0.688·19-s + 4.02·20-s − 1.30·21-s + 2.08·23-s + 4.08·24-s + 6/5·25-s − 2.35·26-s + 1.15·27-s − 3.40·28-s + 1.11·29-s + 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(309372.\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(58.16662396\)
\(L(\frac12)\) \(\approx\) \(58.16662396\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 - 2 T + 2 T^{3} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 20 T^{2} + 54 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + T + 41 T^{2} + 38 T^{3} + 41 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 T + 20 T^{2} - 163 T^{3} + 20 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 4 p T^{2} - 470 T^{3} + 4 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 + 6 T + 65 T^{2} + 388 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 103 T^{2} - 580 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 14 T + 159 T^{2} + 1068 T^{3} + 159 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 7 T + 135 T^{2} + 598 T^{3} + 135 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 6 T + 113 T^{2} - 404 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 9 T + 93 T^{2} - 902 T^{3} + 93 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 11 T + 116 T^{2} - 643 T^{3} + 116 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 3 T + 131 T^{2} + 370 T^{3} + 131 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 19 T + 311 T^{2} - 2742 T^{3} + 311 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 17 T + 233 T^{2} - 2398 T^{3} + 233 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 7 T + 225 T^{2} - 1018 T^{3} + 225 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T + 224 T^{2} + 1417 T^{3} + 224 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 244 T^{2} - 660 T^{3} + 244 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 239 T^{2} - 1084 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 5 T + 131 T^{2} - 1518 T^{3} + 131 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.81898185348300023123208352977, −6.72008652431505778999957942567, −6.37333130429951713446547988452, −6.27954092469827060366360430429, −5.81541156291993325210437001180, −5.65508667957368090107541290007, −5.45320795559727914529097579356, −4.99597607748053004288916937983, −4.98167215629574662728184175044, −4.94589836477319770226985452429, −4.67725694195606627204744319315, −4.05277052576165077742088783641, −4.02620202763772417276898441369, −3.66993567305326112179020337661, −3.50780969922096201518249327680, −3.29681188300595577881242766488, −2.82178497468372408930660903726, −2.69919525201860683473154942473, −2.67066530771975070837000507294, −2.14830962453306976175978867576, −2.13382650904045758320628807022, −1.71558816597654519640782976052, −1.24318051915467954352168827595, −0.947389074494029865234113993451, −0.57705017119737752831639128591, 0.57705017119737752831639128591, 0.947389074494029865234113993451, 1.24318051915467954352168827595, 1.71558816597654519640782976052, 2.13382650904045758320628807022, 2.14830962453306976175978867576, 2.67066530771975070837000507294, 2.69919525201860683473154942473, 2.82178497468372408930660903726, 3.29681188300595577881242766488, 3.50780969922096201518249327680, 3.66993567305326112179020337661, 4.02620202763772417276898441369, 4.05277052576165077742088783641, 4.67725694195606627204744319315, 4.94589836477319770226985452429, 4.98167215629574662728184175044, 4.99597607748053004288916937983, 5.45320795559727914529097579356, 5.65508667957368090107541290007, 5.81541156291993325210437001180, 6.27954092469827060366360430429, 6.37333130429951713446547988452, 6.72008652431505778999957942567, 6.81898185348300023123208352977

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