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Properties

Label 6-38e3-1.1-c9e3-0-0
Degree $6$
Conductor $54872$
Sign $-1$
Analytic cond. $7496.57$
Root an. cond. $4.42395$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 48·2-s + 3·3-s + 1.53e3·4-s + 486·5-s − 144·6-s − 1.33e4·7-s − 4.09e4·8-s − 1.80e4·9-s − 2.33e4·10-s + 5.59e4·11-s + 4.60e3·12-s + 1.58e5·13-s + 6.39e5·14-s + 1.45e3·15-s + 9.83e5·16-s − 6.29e5·17-s + 8.67e5·18-s − 3.90e5·19-s + 7.46e5·20-s − 3.99e4·21-s − 2.68e6·22-s − 9.24e5·23-s − 1.22e5·24-s − 5.21e6·25-s − 7.59e6·26-s − 2.31e6·27-s − 2.04e7·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 0.0213·3-s + 3·4-s + 0.347·5-s − 0.0453·6-s − 2.09·7-s − 3.53·8-s − 0.918·9-s − 0.737·10-s + 1.15·11-s + 0.0641·12-s + 1.53·13-s + 4.44·14-s + 0.00743·15-s + 15/4·16-s − 1.82·17-s + 1.94·18-s − 0.688·19-s + 1.04·20-s − 0.0448·21-s − 2.44·22-s − 0.688·23-s − 0.0756·24-s − 2.66·25-s − 3.25·26-s − 0.836·27-s − 6.28·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54872 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54872 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(54872\)    =    \(2^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(7496.57\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 54872,\ (\ :9/2, 9/2, 9/2),\ -1)\)

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)$
bad2$C_1$ \( ( 1 + p^{4} T )^{3} \)
19$C_1$ \( ( 1 + p^{4} T )^{3} \)
good3$S_4\times C_2$ \( 1 - p T + 2009 p^{2} T^{2} + 81574 p^{3} T^{3} + 2009 p^{11} T^{4} - p^{19} T^{5} + p^{27} T^{6} \)
5$S_4\times C_2$ \( 1 - 486 T + 5450544 T^{2} - 1910152084 T^{3} + 5450544 p^{9} T^{4} - 486 p^{18} T^{5} + p^{27} T^{6} \)
7$S_4\times C_2$ \( 1 + 13317 T + 2817432 p^{2} T^{2} + 20382956301 p^{2} T^{3} + 2817432 p^{11} T^{4} + 13317 p^{18} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 - 5088 p T + 162303954 p T^{2} - 110373793234486 T^{3} + 162303954 p^{10} T^{4} - 5088 p^{19} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 - 158181 T + 36345696879 T^{2} - 3315103030674770 T^{3} + 36345696879 p^{9} T^{4} - 158181 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 + 629091 T + 322401568650 T^{2} + 106985872063124811 T^{3} + 322401568650 p^{9} T^{4} + 629091 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 + 924627 T + 42546695811 p T^{2} + 526819541184348842 T^{3} + 42546695811 p^{10} T^{4} + 924627 p^{18} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 + 9839019 T + 64313088760239 T^{2} + \)\(29\!\cdots\!62\)\( T^{3} + 64313088760239 p^{9} T^{4} + 9839019 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 - 1364628 T + 6528423117837 T^{2} - \)\(10\!\cdots\!36\)\( T^{3} + 6528423117837 p^{9} T^{4} - 1364628 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 + 2289090 T + 309609164810043 T^{2} + \)\(75\!\cdots\!08\)\( T^{3} + 309609164810043 p^{9} T^{4} + 2289090 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 + 12899580 T + 933941039944047 T^{2} + \)\(77\!\cdots\!84\)\( T^{3} + 933941039944047 p^{9} T^{4} + 12899580 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 + 22378638 T + 1149456564786162 T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + 1149456564786162 p^{9} T^{4} + 22378638 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 - 58896366 T + 3778439737379502 T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + 3778439737379502 p^{9} T^{4} - 58896366 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 - 8770629 T + 2855843385985791 T^{2} + \)\(18\!\cdots\!78\)\( T^{3} + 2855843385985791 p^{9} T^{4} - 8770629 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 - 16426299 T + 24576394162259001 T^{2} - \)\(43\!\cdots\!10\)\( p T^{3} + 24576394162259001 p^{9} T^{4} - 16426299 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 + 126843780 T + 39342501567258564 T^{2} + \)\(30\!\cdots\!18\)\( T^{3} + 39342501567258564 p^{9} T^{4} + 126843780 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 + 288075309 T + 92688504651310905 T^{2} + \)\(15\!\cdots\!54\)\( T^{3} + 92688504651310905 p^{9} T^{4} + 288075309 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 - 78122274 T + 123167736240843729 T^{2} - \)\(73\!\cdots\!48\)\( T^{3} + 123167736240843729 p^{9} T^{4} - 78122274 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 + 557941845 T + 181729649630455470 T^{2} + \)\(43\!\cdots\!53\)\( T^{3} + 181729649630455470 p^{9} T^{4} + 557941845 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 320222022 T + 382092522680784861 T^{2} + \)\(76\!\cdots\!36\)\( T^{3} + 382092522680784861 p^{9} T^{4} + 320222022 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 - 430491462 T + 580489855508328153 T^{2} - \)\(15\!\cdots\!28\)\( T^{3} + 580489855508328153 p^{9} T^{4} - 430491462 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 - 437689644 T + 775511286211782651 T^{2} - \)\(33\!\cdots\!72\)\( T^{3} + 775511286211782651 p^{9} T^{4} - 437689644 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 + 384952146 T + 823894073491956771 T^{2} + \)\(18\!\cdots\!72\)\( T^{3} + 823894073491956771 p^{9} T^{4} + 384952146 p^{18} T^{5} + p^{27} T^{6} \)
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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

−13.39515072472408536760867379554, −13.03437298482226944959935300405, −12.20966176231991608067352030159, −11.91639737684654766117965124677, −11.36916345442289835760265903972, −11.20759619341109734133043041513, −10.78314461062718002563688375565, −10.10855420687804206229247804109, −9.758615537894340220968745571513, −9.569435734525768773369944604413, −8.919503424552775472416316832133, −8.811619999593939704645488599172, −8.558661561098450850884668100867, −7.56615790628752382989532129469, −7.42175311837615412091313187202, −6.63715117599022358110881392637, −6.20437660695332779350420407240, −5.98839740348700687620537462371, −5.83641717365814929891329289892, −4.12721804627354224326873669560, −3.65208497007181374866040355129, −3.27973969057249383156501442842, −2.27270317233827806396493344458, −1.92433902697934951331337668580, −1.35808564030174337460759639887, 0, 0, 0, 1.35808564030174337460759639887, 1.92433902697934951331337668580, 2.27270317233827806396493344458, 3.27973969057249383156501442842, 3.65208497007181374866040355129, 4.12721804627354224326873669560, 5.83641717365814929891329289892, 5.98839740348700687620537462371, 6.20437660695332779350420407240, 6.63715117599022358110881392637, 7.42175311837615412091313187202, 7.56615790628752382989532129469, 8.558661561098450850884668100867, 8.811619999593939704645488599172, 8.919503424552775472416316832133, 9.569435734525768773369944604413, 9.758615537894340220968745571513, 10.10855420687804206229247804109, 10.78314461062718002563688375565, 11.20759619341109734133043041513, 11.36916345442289835760265903972, 11.91639737684654766117965124677, 12.20966176231991608067352030159, 13.03437298482226944959935300405, 13.39515072472408536760867379554

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