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Properties

Label 6-63e3-1.1-c9e3-0-1
Degree $6$
Conductor $250047$
Sign $-1$
Analytic cond. $34161.2$
Root an. cond. $5.69624$
Motivic weight $9$
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  − 21·2-s + 231·4-s − 1.55e3·5-s + 7.20e3·7-s − 2.86e3·8-s + 3.26e4·10-s + 3.44e3·11-s − 1.97e4·13-s − 1.51e5·14-s − 4.49e4·16-s − 1.01e6·17-s + 2.22e5·19-s − 3.58e5·20-s − 7.23e4·22-s − 1.88e6·23-s − 1.85e5·25-s + 4.15e5·26-s + 1.66e6·28-s − 4.08e6·29-s + 2.86e6·31-s − 2.26e6·32-s + 2.13e7·34-s − 1.11e7·35-s + 1.39e6·37-s − 4.67e6·38-s + 4.45e6·40-s + 1.44e7·41-s + ⋯
L(s)  = 1  − 0.928·2-s + 0.451·4-s − 1.11·5-s + 1.13·7-s − 0.247·8-s + 1.03·10-s + 0.0709·11-s − 0.192·13-s − 1.05·14-s − 0.171·16-s − 2.95·17-s + 0.392·19-s − 0.501·20-s − 0.0658·22-s − 1.40·23-s − 0.0950·25-s + 0.178·26-s + 0.511·28-s − 1.07·29-s + 0.558·31-s − 0.381·32-s + 2.74·34-s − 1.26·35-s + 0.122·37-s − 0.364·38-s + 0.274·40-s + 0.796·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250047 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250047 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(250047\)    =    \(3^{6} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(34161.2\)
Root analytic conductor: \(5.69624\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 250047,\ (\ :9/2, 9/2, 9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{4} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 21 T + 105 p T^{2} + 303 p^{3} T^{3} + 105 p^{10} T^{4} + 21 p^{18} T^{5} + p^{27} T^{6} \)
5$S_4\times C_2$ \( 1 + 1554 T + 520107 p T^{2} + 46663764 p^{2} T^{3} + 520107 p^{10} T^{4} + 1554 p^{18} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 - 3444 T + 455343105 T^{2} - 125101303155960 T^{3} + 455343105 p^{9} T^{4} - 3444 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 + 19782 T + 18882215055 T^{2} + 378008000651932 T^{3} + 18882215055 p^{9} T^{4} + 19782 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 + 1016694 T + 649258140783 T^{2} + 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} + 1016694 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 - 222852 T + 614081373717 T^{2} - 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} - 222852 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 + 81984 p T + 5417652680517 T^{2} + 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} + 81984 p^{19} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 + 4081818 T + 38739015783987 T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} + 4081818 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 - 2869440 T + 21142500166221 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} - 2869440 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 - 1395618 T + 262675972194027 T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + 262675972194027 p^{9} T^{4} - 1395618 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 - 14420658 T + 764979654799959 T^{2} - \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} - 14420658 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 + 61631172 T + 2687603003165025 T^{2} + \)\(16\!\cdots\!04\)\( p T^{3} + 2687603003165025 p^{9} T^{4} + 61631172 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 - 10368960 T + 2946826961339709 T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + 2946826961339709 p^{9} T^{4} - 10368960 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 + 67502610 T + 6295046710287531 T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + 6295046710287531 p^{9} T^{4} + 67502610 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 - 42590100 T + 19076976504365997 T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} - 42590100 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 - 191746842 T + 38596678668907359 T^{2} - \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 + 255175788 T + 80172518705654361 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + 80172518705654361 p^{9} T^{4} + 255175788 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 + 296514504 T + 147895725194380437 T^{2} + \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} + 296514504 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 - 344213310 T + 83298340302082311 T^{2} - \)\(20\!\cdots\!12\)\( T^{3} + 83298340302082311 p^{9} T^{4} - 344213310 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 960412656 T + 566786434394061357 T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} + 960412656 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 - 1100517180 T + 873896700882341301 T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + 873896700882341301 p^{9} T^{4} - 1100517180 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 + 506816478 T + 956194525794688887 T^{2} + \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} + 506816478 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 + 647498250 T + 819696244591424799 T^{2} + \)\(48\!\cdots\!84\)\( T^{3} + 819696244591424799 p^{9} T^{4} + 647498250 p^{18} T^{5} + p^{27} T^{6} \)
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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

−12.07185482586371829030259371578, −11.47651951040733311141169862289, −11.42397333984573649373725490476, −10.93487471758985842688231556022, −10.86420414262087081229650812908, −10.04894910797532452527094488332, −9.859955405574558060006673041690, −9.264742656435159833075878388496, −8.760110974183384334343837870780, −8.714252936357873545475765098546, −8.056491474823837420230041668973, −7.928447044285471138001362957172, −7.46235458863671044201541615440, −6.84595703704869970330207207381, −6.71042080146686113068152406349, −6.02599315580771115065025801236, −5.45196070994427558815889284418, −4.72079162221499812204051788963, −4.58259433621368708441529712567, −3.80641240147590263003835847871, −3.77246288940839410234400663429, −2.49319138010226737124441985149, −2.38843074118005536910573185663, −1.59471478908500142934359426049, −1.30422893311466732309819377029, 0, 0, 0, 1.30422893311466732309819377029, 1.59471478908500142934359426049, 2.38843074118005536910573185663, 2.49319138010226737124441985149, 3.77246288940839410234400663429, 3.80641240147590263003835847871, 4.58259433621368708441529712567, 4.72079162221499812204051788963, 5.45196070994427558815889284418, 6.02599315580771115065025801236, 6.71042080146686113068152406349, 6.84595703704869970330207207381, 7.46235458863671044201541615440, 7.928447044285471138001362957172, 8.056491474823837420230041668973, 8.714252936357873545475765098546, 8.760110974183384334343837870780, 9.264742656435159833075878388496, 9.859955405574558060006673041690, 10.04894910797532452527094488332, 10.86420414262087081229650812908, 10.93487471758985842688231556022, 11.42397333984573649373725490476, 11.47651951040733311141169862289, 12.07185482586371829030259371578

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