Properties

Label 6-5070e3-1.1-c1e3-0-1
Degree $6$
Conductor $130323843000$
Sign $1$
Analytic cond. $66352.1$
Root an. cond. $6.36271$
Motivic weight $1$
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  − 3·2-s − 3·3-s + 6·4-s − 3·5-s + 9·6-s − 10·8-s + 6·9-s + 9·10-s + 14·11-s − 18·12-s + 9·15-s + 15·16-s + 3·17-s − 18·18-s + 7·19-s − 18·20-s − 42·22-s − 5·23-s + 30·24-s + 6·25-s − 10·27-s + 12·29-s − 27·30-s + 14·31-s − 21·32-s − 42·33-s − 9·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s − 3.53·8-s + 2·9-s + 2.84·10-s + 4.22·11-s − 5.19·12-s + 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s + 1.60·19-s − 4.02·20-s − 8.95·22-s − 1.04·23-s + 6.12·24-s + 6/5·25-s − 1.92·27-s + 2.22·29-s − 4.92·30-s + 2.51·31-s − 3.71·32-s − 7.31·33-s − 1.54·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(66352.1\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.374249735\)
\(L(\frac12)\) \(\approx\) \(1.374249735\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good7$A_4\times C_2$ \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 14 T + 96 T^{2} - 399 T^{3} + 96 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 3 T + 47 T^{2} - 89 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 7 T + 29 T^{2} - 63 T^{3} + 29 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 5 T - 9 T^{2} - 189 T^{3} - 9 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 11 p T^{3} + 72 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 14 T + 128 T^{2} - 861 T^{3} + 128 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 4 T + 86 T^{2} + 225 T^{3} + 86 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 13 T + 121 T^{2} - 927 T^{3} + 121 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 5 T + 121 T^{2} + 389 T^{3} + 121 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 21 T + 260 T^{2} - 2177 T^{3} + 260 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 3 T + 141 T^{2} - 305 T^{3} + 141 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 7 T + 107 T^{2} - 917 T^{3} + 107 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 17 T + 179 T^{2} + 1473 T^{3} + 179 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 5 T + 207 T^{2} - 671 T^{3} + 207 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
71$C_6$ \( 1 - 13 T + 183 T^{2} - 1833 T^{3} + 183 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 13 T + 161 T^{2} + 1395 T^{3} + 161 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 7 T + 181 T^{2} + 1015 T^{3} + 181 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 12 T + 206 T^{2} - 1405 T^{3} + 206 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 3 T + 179 T^{2} - 241 T^{3} + 179 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 270 T^{2} + 7 T^{3} + 270 p T^{4} + p^{3} T^{6} \)
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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

−7.31075237809650076807888570509, −6.98665011183212563956694703331, −6.71307004264513881874196637460, −6.66575327570663301926197889298, −6.48358192850996901928129779511, −6.31917177135835036275046433639, −6.01678383167122941514830628513, −5.78257793118141653121458651401, −5.46451343059103464595566188802, −5.13650691792435437914904068013, −4.80660490673680163759695187625, −4.32964040596805463641252783431, −4.27401239050428083200440962019, −4.06629924642997500877030738959, −3.72314610947989989444636360494, −3.61970400401836408871970337608, −2.91737397049297718024868799860, −2.87386017577320539823888137076, −2.59255091548729693927150326837, −1.69617673626020252369096680466, −1.51440730829124522439258797641, −1.33674256028349186237311815988, −0.947557516638854379840405046474, −0.64740718895793163087470704205, −0.61614507003583976689337394429, 0.61614507003583976689337394429, 0.64740718895793163087470704205, 0.947557516638854379840405046474, 1.33674256028349186237311815988, 1.51440730829124522439258797641, 1.69617673626020252369096680466, 2.59255091548729693927150326837, 2.87386017577320539823888137076, 2.91737397049297718024868799860, 3.61970400401836408871970337608, 3.72314610947989989444636360494, 4.06629924642997500877030738959, 4.27401239050428083200440962019, 4.32964040596805463641252783431, 4.80660490673680163759695187625, 5.13650691792435437914904068013, 5.46451343059103464595566188802, 5.78257793118141653121458651401, 6.01678383167122941514830628513, 6.31917177135835036275046433639, 6.48358192850996901928129779511, 6.66575327570663301926197889298, 6.71307004264513881874196637460, 6.98665011183212563956694703331, 7.31075237809650076807888570509

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