Properties

Label 6-5586e3-1.1-c1e3-0-1
Degree $6$
Conductor $174302170056$
Sign $1$
Analytic cond. $88743.0$
Root an. cond. $6.67865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 6·4-s + 5·5-s − 9·6-s + 10·8-s + 6·9-s + 15·10-s + 11-s − 18·12-s + 6·13-s − 15·15-s + 15·16-s + 14·17-s + 18·18-s + 3·19-s + 30·20-s + 3·22-s − 6·23-s − 30·24-s + 7·25-s + 18·26-s − 10·27-s − 29-s − 45·30-s + 11·31-s + 21·32-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s + 2.23·5-s − 3.67·6-s + 3.53·8-s + 2·9-s + 4.74·10-s + 0.301·11-s − 5.19·12-s + 1.66·13-s − 3.87·15-s + 15/4·16-s + 3.39·17-s + 4.24·18-s + 0.688·19-s + 6.70·20-s + 0.639·22-s − 1.25·23-s − 6.12·24-s + 7/5·25-s + 3.53·26-s − 1.92·27-s − 0.185·29-s − 8.21·30-s + 1.97·31-s + 3.71·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(88743.0\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.44115006\)
\(L(\frac12)\) \(\approx\) \(34.44115006\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 - p T + 18 T^{2} - 43 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - T + 16 T^{2} - T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 19 T^{2} - 68 T^{3} + 19 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 14 T + 111 T^{2} - 550 T^{3} + 111 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 15 T^{2} - 26 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T + 46 T^{2} + 101 T^{3} + 46 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 11 T + 112 T^{2} - 675 T^{3} + 112 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 67 T^{2} + 102 T^{3} + 67 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 169 T^{2} - 1190 T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 6 T + 37 T^{2} + 90 T^{3} + 37 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 45 T^{2} + 520 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 9 T + 154 T^{2} + 853 T^{3} + 154 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 7 T + 174 T^{2} - 763 T^{3} + 174 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 10 T + 195 T^{2} + 1164 T^{3} + 195 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T + 217 T^{2} + 1576 T^{3} + 217 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 6 T + 121 T^{2} + 1058 T^{3} + 121 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 2 T + 163 T^{2} - 96 T^{3} + 163 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 15 T + 156 T^{2} + 1303 T^{3} + 156 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 19 T + 92 T^{2} + 125 T^{3} + 92 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 313 T^{2} - 2534 T^{3} + 313 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 31 T + 592 T^{2} - 71 p T^{3} + 592 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.06845017342903574025608037188, −6.44907709166868660523448902196, −6.43345110173128618426511835072, −6.33023872643963382554733072488, −6.02598001685967625240060159979, −5.94109572242547238469772137122, −5.84478985772312388357986391752, −5.49183549026942927962084803838, −5.36500378106148952262980672603, −5.29782528447603543091148784463, −4.66990158976984762999655461203, −4.62223479326900680077673913678, −4.49354634821691882278693327108, −3.82889123305250956997715929236, −3.79310300293442551745935927816, −3.59631260478483836067179617810, −3.12769954940587944187514763924, −2.94749447739596914339067043663, −2.67843931822768327442984127021, −2.00688629947270061415009515500, −1.99620292612982402555272660190, −1.53517168472584722236199304705, −1.27303734125372128739419814807, −1.01649785618774800150890893045, −0.70553034197051503895251733462, 0.70553034197051503895251733462, 1.01649785618774800150890893045, 1.27303734125372128739419814807, 1.53517168472584722236199304705, 1.99620292612982402555272660190, 2.00688629947270061415009515500, 2.67843931822768327442984127021, 2.94749447739596914339067043663, 3.12769954940587944187514763924, 3.59631260478483836067179617810, 3.79310300293442551745935927816, 3.82889123305250956997715929236, 4.49354634821691882278693327108, 4.62223479326900680077673913678, 4.66990158976984762999655461203, 5.29782528447603543091148784463, 5.36500378106148952262980672603, 5.49183549026942927962084803838, 5.84478985772312388357986391752, 5.94109572242547238469772137122, 6.02598001685967625240060159979, 6.33023872643963382554733072488, 6.43345110173128618426511835072, 6.44907709166868660523448902196, 7.06845017342903574025608037188

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