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Properties

Label 6-7e6-1.1-c13e3-0-0
Degree $6$
Conductor $117649$
Sign $1$
Analytic cond. $145060.$
Root an. cond. $7.24866$
Motivic weight $13$
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  + 26·2-s + 1.79e3·3-s − 3.12e3·4-s + 2.40e4·5-s + 4.66e4·6-s − 3.44e5·8-s − 8.46e5·9-s + 6.26e5·10-s − 9.33e6·11-s − 5.61e6·12-s − 9.21e6·13-s + 4.32e7·15-s − 3.25e7·16-s + 1.61e8·17-s − 2.20e7·18-s − 3.51e8·19-s − 7.52e7·20-s − 2.42e8·22-s − 1.25e9·23-s − 6.19e8·24-s − 1.67e9·25-s − 2.39e8·26-s − 3.12e9·27-s − 6.74e9·29-s + 1.12e9·30-s − 2.96e9·31-s + 1.91e9·32-s + ⋯
L(s)  = 1  + 0.287·2-s + 1.42·3-s − 0.381·4-s + 0.689·5-s + 0.408·6-s − 0.464·8-s − 0.531·9-s + 0.198·10-s − 1.58·11-s − 0.542·12-s − 0.529·13-s + 0.980·15-s − 0.484·16-s + 1.62·17-s − 0.152·18-s − 1.71·19-s − 0.262·20-s − 0.456·22-s − 1.77·23-s − 0.661·24-s − 1.37·25-s − 0.152·26-s − 1.55·27-s − 2.10·29-s + 0.281·30-s − 0.599·31-s + 0.315·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117649 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117649 ^{s/2} \, \Gamma_{\C}(s+13/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(117649\)    =    \(7^{6}\)
Sign: $1$
Analytic conductor: \(145060.\)
Root analytic conductor: \(7.24866\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 117649,\ (\ :13/2, 13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.992315526\)
\(L(\frac12)\) \(\approx\) \(1.992315526\)
\(L(\frac{15}{2})\) 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)$
bad7 \( 1 \)
good2$S_4\times C_2$ \( 1 - 13 p T + 475 p^{3} T^{2} + 1287 p^{7} T^{3} + 475 p^{16} T^{4} - 13 p^{27} T^{5} + p^{39} T^{6} \)
3$S_4\times C_2$ \( 1 - 1796 T + 1357471 p T^{2} - 70439480 p^{4} T^{3} + 1357471 p^{14} T^{4} - 1796 p^{26} T^{5} + p^{39} T^{6} \)
5$S_4\times C_2$ \( 1 - 24086 T + 450627859 p T^{2} - 1716598818684 p^{2} T^{3} + 450627859 p^{14} T^{4} - 24086 p^{26} T^{5} + p^{39} T^{6} \)
11$S_4\times C_2$ \( 1 + 849004 p T + 1090750862585 p^{2} T^{2} + 505730266289181960 p^{3} T^{3} + 1090750862585 p^{15} T^{4} + 849004 p^{27} T^{5} + p^{39} T^{6} \)
13$S_4\times C_2$ \( 1 + 9219378 T + 742226686518615 T^{2} + \)\(39\!\cdots\!88\)\( T^{3} + 742226686518615 p^{13} T^{4} + 9219378 p^{26} T^{5} + p^{39} T^{6} \)
17$S_4\times C_2$ \( 1 - 161799306 T + 6122252444646543 T^{2} + \)\(61\!\cdots\!08\)\( T^{3} + 6122252444646543 p^{13} T^{4} - 161799306 p^{26} T^{5} + p^{39} T^{6} \)
19$S_4\times C_2$ \( 1 + 351536172 T + 135665600505753837 T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + 135665600505753837 p^{13} T^{4} + 351536172 p^{26} T^{5} + p^{39} T^{6} \)
23$S_4\times C_2$ \( 1 + 1258991568 T + 1768220730616582437 T^{2} + \)\(12\!\cdots\!84\)\( T^{3} + 1768220730616582437 p^{13} T^{4} + 1258991568 p^{26} T^{5} + p^{39} T^{6} \)
29$S_4\times C_2$ \( 1 + 6748418342 T + 37165622688043242947 T^{2} + \)\(12\!\cdots\!76\)\( T^{3} + 37165622688043242947 p^{13} T^{4} + 6748418342 p^{26} T^{5} + p^{39} T^{6} \)
31$S_4\times C_2$ \( 1 + 2961621120 T + 25205383159005864621 T^{2} + \)\(22\!\cdots\!44\)\( T^{3} + 25205383159005864621 p^{13} T^{4} + 2961621120 p^{26} T^{5} + p^{39} T^{6} \)
37$S_4\times C_2$ \( 1 + 15165028062 T + \)\(70\!\cdots\!87\)\( T^{2} + \)\(69\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!87\)\( p^{13} T^{4} + 15165028062 p^{26} T^{5} + p^{39} T^{6} \)
41$S_4\times C_2$ \( 1 - 30348543778 T + \)\(15\!\cdots\!99\)\( T^{2} - \)\(32\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!99\)\( p^{13} T^{4} - 30348543778 p^{26} T^{5} + p^{39} T^{6} \)
43$S_4\times C_2$ \( 1 + 80250536052 T + \)\(67\!\cdots\!25\)\( T^{2} + \)\(27\!\cdots\!12\)\( T^{3} + \)\(67\!\cdots\!25\)\( p^{13} T^{4} + 80250536052 p^{26} T^{5} + p^{39} T^{6} \)
47$S_4\times C_2$ \( 1 - 169583042880 T + \)\(20\!\cdots\!09\)\( T^{2} - \)\(16\!\cdots\!64\)\( T^{3} + \)\(20\!\cdots\!09\)\( p^{13} T^{4} - 169583042880 p^{26} T^{5} + p^{39} T^{6} \)
53$S_4\times C_2$ \( 1 - 120814398690 T + \)\(58\!\cdots\!71\)\( T^{2} - \)\(45\!\cdots\!32\)\( T^{3} + \)\(58\!\cdots\!71\)\( p^{13} T^{4} - 120814398690 p^{26} T^{5} + p^{39} T^{6} \)
59$S_4\times C_2$ \( 1 + 443036517780 T + \)\(33\!\cdots\!57\)\( T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!57\)\( p^{13} T^{4} + 443036517780 p^{26} T^{5} + p^{39} T^{6} \)
61$S_4\times C_2$ \( 1 - 312164967918 T + \)\(18\!\cdots\!79\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!79\)\( p^{13} T^{4} - 312164967918 p^{26} T^{5} + p^{39} T^{6} \)
67$S_4\times C_2$ \( 1 - 1398804629172 T + \)\(19\!\cdots\!81\)\( T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!81\)\( p^{13} T^{4} - 1398804629172 p^{26} T^{5} + p^{39} T^{6} \)
71$S_4\times C_2$ \( 1 - 980341716024 T + \)\(13\!\cdots\!37\)\( T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!37\)\( p^{13} T^{4} - 980341716024 p^{26} T^{5} + p^{39} T^{6} \)
73$S_4\times C_2$ \( 1 + 964409395470 T + \)\(18\!\cdots\!51\)\( T^{2} + \)\(22\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!51\)\( p^{13} T^{4} + 964409395470 p^{26} T^{5} + p^{39} T^{6} \)
79$S_4\times C_2$ \( 1 - 5421189462624 T + \)\(18\!\cdots\!97\)\( T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!97\)\( p^{13} T^{4} - 5421189462624 p^{26} T^{5} + p^{39} T^{6} \)
83$S_4\times C_2$ \( 1 - 5151799373700 T + \)\(29\!\cdots\!01\)\( T^{2} - \)\(81\!\cdots\!28\)\( T^{3} + \)\(29\!\cdots\!01\)\( p^{13} T^{4} - 5151799373700 p^{26} T^{5} + p^{39} T^{6} \)
89$S_4\times C_2$ \( 1 - 15420753518162 T + \)\(13\!\cdots\!67\)\( T^{2} - \)\(78\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!67\)\( p^{13} T^{4} - 15420753518162 p^{26} T^{5} + p^{39} T^{6} \)
97$S_4\times C_2$ \( 1 + 4897266340470 T + \)\(20\!\cdots\!59\)\( T^{2} + \)\(64\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!59\)\( p^{13} T^{4} + 4897266340470 p^{26} T^{5} + p^{39} T^{6} \)
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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

−11.37642850272642235632849684525, −10.52666704715667613087632259058, −10.31083395141857782878798883670, −10.28163715025435615095423569907, −9.301367092586496977662495683899, −9.153783976664409774878545885362, −9.124897448607123702801801560815, −8.225965210172948293050690284151, −7.956975194062650279910304748725, −7.86265942666514572191172417610, −7.44824351448550709231144839757, −6.35270976487075147692375130463, −6.29381886004141218315814752941, −5.56911693213658228016703720845, −5.29843484044687486682599822470, −5.09314696063247641496934175048, −4.03925000812581661055916434647, −3.69832584128973832027607405928, −3.50062443872246350725414171338, −2.60317834688607186504340705453, −2.49692282013314354517392080665, −1.94323109436925294593239727734, −1.89224964470895300229919537595, −0.54256677720375416150295989526, −0.27594195993483753163734674959, 0.27594195993483753163734674959, 0.54256677720375416150295989526, 1.89224964470895300229919537595, 1.94323109436925294593239727734, 2.49692282013314354517392080665, 2.60317834688607186504340705453, 3.50062443872246350725414171338, 3.69832584128973832027607405928, 4.03925000812581661055916434647, 5.09314696063247641496934175048, 5.29843484044687486682599822470, 5.56911693213658228016703720845, 6.29381886004141218315814752941, 6.35270976487075147692375130463, 7.44824351448550709231144839757, 7.86265942666514572191172417610, 7.956975194062650279910304748725, 8.225965210172948293050690284151, 9.124897448607123702801801560815, 9.153783976664409774878545885362, 9.301367092586496977662495683899, 10.28163715025435615095423569907, 10.31083395141857782878798883670, 10.52666704715667613087632259058, 11.37642850272642235632849684525

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