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Properties

Label 6-75e3-1.1-c21e3-0-1
Degree $6$
Conductor $421875$
Sign $-1$
Analytic cond. $9.20923\times 10^{6}$
Root an. cond. $14.4778$
Motivic weight $21$
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  − 803·2-s − 1.77e5·3-s + 5.34e5·4-s + 1.42e8·6-s + 1.57e9·7-s + 7.79e8·8-s + 2.09e10·9-s + 8.44e10·11-s − 9.46e10·12-s − 1.06e12·13-s − 1.26e12·14-s − 1.81e12·16-s − 1.38e13·17-s − 1.67e13·18-s + 2.68e13·19-s − 2.79e14·21-s − 6.78e13·22-s − 7.57e13·23-s − 1.38e14·24-s + 8.55e14·26-s − 2.05e15·27-s + 8.42e14·28-s − 9.11e14·29-s − 1.47e16·31-s − 1.37e15·32-s − 1.49e16·33-s + 1.11e16·34-s + ⋯
L(s)  = 1  − 0.554·2-s − 1.73·3-s + 0.254·4-s + 0.960·6-s + 2.11·7-s + 0.256·8-s + 2·9-s + 0.982·11-s − 0.441·12-s − 2.14·13-s − 1.17·14-s − 0.412·16-s − 1.66·17-s − 1.10·18-s + 1.00·19-s − 3.65·21-s − 0.544·22-s − 0.381·23-s − 0.444·24-s + 1.18·26-s − 1.92·27-s + 0.537·28-s − 0.402·29-s − 3.23·31-s − 0.215·32-s − 1.70·33-s + 0.924·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(421875\)    =    \(3^{3} \cdot 5^{6}\)
Sign: $-1$
Analytic conductor: \(9.20923\times 10^{6}\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 421875,\ (\ :21/2, 21/2, 21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 + p^{10} T )^{3} \)
5 \( 1 \)
good2$S_4\times C_2$ \( 1 + 803 T + 27649 p^{2} T^{2} - 4374427 p^{8} T^{3} + 27649 p^{23} T^{4} + 803 p^{42} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 4599412 p^{3} T + 32558376418555125 p^{2} T^{2} - \)\(40\!\cdots\!08\)\( p^{3} T^{3} + 32558376418555125 p^{23} T^{4} - 4599412 p^{45} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 - 84497282000 T + \)\(21\!\cdots\!51\)\( p T^{2} - \)\(10\!\cdots\!84\)\( p^{2} T^{3} + \)\(21\!\cdots\!51\)\( p^{22} T^{4} - 84497282000 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 + 1065489966310 T + \)\(74\!\cdots\!27\)\( p T^{2} + \)\(31\!\cdots\!52\)\( p^{2} T^{3} + \)\(74\!\cdots\!27\)\( p^{22} T^{4} + 1065489966310 p^{42} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 + 13851876239906 T + \)\(14\!\cdots\!39\)\( p T^{2} + \)\(65\!\cdots\!08\)\( p^{2} T^{3} + \)\(14\!\cdots\!39\)\( p^{22} T^{4} + 13851876239906 p^{42} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 - 26858848298644 T + \)\(81\!\cdots\!43\)\( p T^{2} + \)\(42\!\cdots\!48\)\( p^{2} T^{3} + \)\(81\!\cdots\!43\)\( p^{22} T^{4} - 26858848298644 p^{42} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 + 75776598293952 T + \)\(11\!\cdots\!65\)\( T^{2} + \)\(58\!\cdots\!12\)\( T^{3} + \)\(11\!\cdots\!65\)\( p^{21} T^{4} + 75776598293952 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 + 911172744177122 T + \)\(35\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!07\)\( p^{21} T^{4} + 911172744177122 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 + 14770635893644696 T + \)\(13\!\cdots\!97\)\( T^{2} + \)\(71\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!97\)\( p^{21} T^{4} + 14770635893644696 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 - 33876604648537146 T + \)\(26\!\cdots\!95\)\( T^{2} - \)\(56\!\cdots\!44\)\( T^{3} + \)\(26\!\cdots\!95\)\( p^{21} T^{4} - 33876604648537146 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 - 4403226550677934 T - \)\(24\!\cdots\!13\)\( T^{2} - \)\(93\!\cdots\!88\)\( T^{3} - \)\(24\!\cdots\!13\)\( p^{21} T^{4} - 4403226550677934 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 - 76568115811359884 T + \)\(30\!\cdots\!73\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!73\)\( p^{21} T^{4} - 76568115811359884 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 + 178877159689418912 T + \)\(14\!\cdots\!77\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!77\)\( p^{21} T^{4} + 178877159689418912 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 + 4013622553495571102 T + \)\(83\!\cdots\!95\)\( T^{2} + \)\(12\!\cdots\!72\)\( T^{3} + \)\(83\!\cdots\!95\)\( p^{21} T^{4} + 4013622553495571102 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 + 4476302860362619024 T + \)\(24\!\cdots\!77\)\( T^{2} + \)\(70\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!77\)\( p^{21} T^{4} + 4476302860362619024 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 - 13672812736400197042 T + \)\(14\!\cdots\!79\)\( T^{2} - \)\(89\!\cdots\!56\)\( T^{3} + \)\(14\!\cdots\!79\)\( p^{21} T^{4} - 13672812736400197042 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 + 10024292550718281716 T + \)\(44\!\cdots\!53\)\( T^{2} + \)\(47\!\cdots\!92\)\( T^{3} + \)\(44\!\cdots\!53\)\( p^{21} T^{4} + 10024292550718281716 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 31548512648354310016 T + \)\(13\!\cdots\!65\)\( T^{2} - \)\(33\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!65\)\( p^{21} T^{4} - 31548512648354310016 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 + 80637656440064589382 T + \)\(81\!\cdots\!15\)\( p T^{2} + \)\(22\!\cdots\!92\)\( T^{3} + \)\(81\!\cdots\!15\)\( p^{22} T^{4} + 80637656440064589382 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(15\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!37\)\( p^{21} T^{4} + \)\(10\!\cdots\!80\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 - 98643502167172343676 T + \)\(23\!\cdots\!89\)\( T^{2} - \)\(29\!\cdots\!24\)\( T^{3} + \)\(23\!\cdots\!89\)\( p^{21} T^{4} - 98643502167172343676 p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 - 49757751372279077814 T + \)\(74\!\cdots\!27\)\( T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(74\!\cdots\!27\)\( p^{21} T^{4} - 49757751372279077814 p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(76\!\cdots\!66\)\( T - \)\(36\!\cdots\!57\)\( T^{2} - \)\(82\!\cdots\!48\)\( T^{3} - \)\(36\!\cdots\!57\)\( p^{21} T^{4} + \)\(76\!\cdots\!66\)\( p^{42} T^{5} + p^{63} T^{6} \)
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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

−9.859242716278973788015217426737, −9.369105439194281653412679256823, −9.046327626753337592963924058605, −9.004458175072843850519200378538, −8.046314082020998137381658908610, −7.957858299218419687105484165076, −7.46416101167472996620869153367, −7.17386994725067211127780975418, −7.17272662037719855597236803809, −6.44999403846955098669894508598, −6.23842269550892094546688358687, −5.59434726866269566204907544083, −5.42680086157127671860381894331, −4.86059228013340531983625673998, −4.84109855595610076966260885312, −4.54431864979303597850064283837, −4.01398742955303427350417208845, −3.80867625114834502902918182216, −3.02095114450699448904814605772, −2.50170782926276738842947240428, −1.96343175165254905925358599741, −1.77582019282376849635299205786, −1.62063130604367277753000103041, −1.18735772229814110266576035341, −0.76476867949431942694983218637, 0, 0, 0, 0.76476867949431942694983218637, 1.18735772229814110266576035341, 1.62063130604367277753000103041, 1.77582019282376849635299205786, 1.96343175165254905925358599741, 2.50170782926276738842947240428, 3.02095114450699448904814605772, 3.80867625114834502902918182216, 4.01398742955303427350417208845, 4.54431864979303597850064283837, 4.84109855595610076966260885312, 4.86059228013340531983625673998, 5.42680086157127671860381894331, 5.59434726866269566204907544083, 6.23842269550892094546688358687, 6.44999403846955098669894508598, 7.17272662037719855597236803809, 7.17386994725067211127780975418, 7.46416101167472996620869153367, 7.957858299218419687105484165076, 8.046314082020998137381658908610, 9.004458175072843850519200378538, 9.046327626753337592963924058605, 9.369105439194281653412679256823, 9.859242716278973788015217426737

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