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Properties

Label 6-48e3-1.1-c23e3-0-1
Degree $6$
Conductor $110592$
Sign $1$
Analytic cond. $4.16534\times 10^{6}$
Root an. cond. $12.6845$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.31e5·3-s − 4.50e7·5-s − 2.81e8·7-s + 1.88e11·9-s + 1.19e12·11-s − 2.82e11·13-s + 2.39e13·15-s + 1.86e14·17-s + 1.18e14·19-s + 1.49e14·21-s + 6.39e15·23-s − 1.55e16·25-s − 5.55e16·27-s − 1.57e17·29-s + 2.31e17·31-s − 6.37e17·33-s + 1.26e16·35-s − 8.30e17·37-s + 1.50e17·39-s − 1.40e18·41-s − 7.61e17·43-s − 8.47e18·45-s − 2.07e19·47-s − 2.76e19·49-s − 9.93e19·51-s − 6.46e19·53-s − 5.39e19·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.412·5-s − 0.0537·7-s + 2·9-s + 1.26·11-s − 0.0437·13-s + 0.714·15-s + 1.32·17-s + 0.233·19-s + 0.0931·21-s + 1.39·23-s − 1.30·25-s − 1.92·27-s − 2.40·29-s + 1.63·31-s − 2.19·33-s + 0.0221·35-s − 0.767·37-s + 0.0758·39-s − 0.397·41-s − 0.124·43-s − 0.824·45-s − 1.22·47-s − 1.00·49-s − 2.29·51-s − 0.958·53-s − 0.522·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+23/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(110592\)    =    \(2^{12} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(4.16534\times 10^{6}\)
Root analytic conductor: \(12.6845\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 110592,\ (\ :23/2, 23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.9454134485\)
\(L(\frac12)\) \(\approx\) \(0.9454134485\)
\(L(\frac{25}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + p^{11} T )^{3} \)
good5$S_4\times C_2$ \( 1 + 1800486 p^{2} T + 704309084449419 p^{2} T^{2} + \)\(24\!\cdots\!44\)\( p^{5} T^{3} + 704309084449419 p^{25} T^{4} + 1800486 p^{48} T^{5} + p^{69} T^{6} \)
7$S_4\times C_2$ \( 1 + 281292072 T + 565742548348803381 p^{2} T^{2} + \)\(65\!\cdots\!92\)\( p^{4} T^{3} + 565742548348803381 p^{25} T^{4} + 281292072 p^{46} T^{5} + p^{69} T^{6} \)
11$S_4\times C_2$ \( 1 - 108970550652 p T + \)\(21\!\cdots\!39\)\( p^{3} T^{2} - \)\(15\!\cdots\!64\)\( p^{3} T^{3} + \)\(21\!\cdots\!39\)\( p^{26} T^{4} - 108970550652 p^{47} T^{5} + p^{69} T^{6} \)
13$S_4\times C_2$ \( 1 + 282994829454 T + \)\(38\!\cdots\!51\)\( T^{2} - \)\(21\!\cdots\!68\)\( p^{2} T^{3} + \)\(38\!\cdots\!51\)\( p^{23} T^{4} + 282994829454 p^{46} T^{5} + p^{69} T^{6} \)
17$S_4\times C_2$ \( 1 - 186934623333462 T + \)\(17\!\cdots\!83\)\( p^{2} T^{2} - \)\(26\!\cdots\!84\)\( p^{2} T^{3} + \)\(17\!\cdots\!83\)\( p^{25} T^{4} - 186934623333462 p^{46} T^{5} + p^{69} T^{6} \)
19$S_4\times C_2$ \( 1 - 118422337604076 T + \)\(13\!\cdots\!79\)\( p T^{2} - \)\(13\!\cdots\!72\)\( p^{2} T^{3} + \)\(13\!\cdots\!79\)\( p^{24} T^{4} - 118422337604076 p^{46} T^{5} + p^{69} T^{6} \)
23$S_4\times C_2$ \( 1 - 6395792138657544 T + \)\(64\!\cdots\!25\)\( T^{2} - \)\(24\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!25\)\( p^{23} T^{4} - 6395792138657544 p^{46} T^{5} + p^{69} T^{6} \)
29$S_4\times C_2$ \( 1 + 157790889708267966 T + \)\(89\!\cdots\!31\)\( T^{2} + \)\(36\!\cdots\!72\)\( T^{3} + \)\(89\!\cdots\!31\)\( p^{23} T^{4} + 157790889708267966 p^{46} T^{5} + p^{69} T^{6} \)
31$S_4\times C_2$ \( 1 - 231251640669554784 T + \)\(48\!\cdots\!57\)\( T^{2} - \)\(60\!\cdots\!28\)\( T^{3} + \)\(48\!\cdots\!57\)\( p^{23} T^{4} - 231251640669554784 p^{46} T^{5} + p^{69} T^{6} \)
37$S_4\times C_2$ \( 1 + 830491747621421334 T + \)\(19\!\cdots\!11\)\( T^{2} + \)\(57\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!11\)\( p^{23} T^{4} + 830491747621421334 p^{46} T^{5} + p^{69} T^{6} \)
41$S_4\times C_2$ \( 1 + 1400039039679261042 T + \)\(30\!\cdots\!19\)\( T^{2} + \)\(29\!\cdots\!04\)\( T^{3} + \)\(30\!\cdots\!19\)\( p^{23} T^{4} + 1400039039679261042 p^{46} T^{5} + p^{69} T^{6} \)
43$S_4\times C_2$ \( 1 + 761384577026785356 T + \)\(65\!\cdots\!33\)\( T^{2} + \)\(94\!\cdots\!92\)\( T^{3} + \)\(65\!\cdots\!33\)\( p^{23} T^{4} + 761384577026785356 p^{46} T^{5} + p^{69} T^{6} \)
47$S_4\times C_2$ \( 1 + 20706191933413048944 T + \)\(49\!\cdots\!29\)\( T^{2} + \)\(48\!\cdots\!24\)\( T^{3} + \)\(49\!\cdots\!29\)\( p^{23} T^{4} + 20706191933413048944 p^{46} T^{5} + p^{69} T^{6} \)
53$S_4\times C_2$ \( 1 + 64674463906638102342 T + \)\(86\!\cdots\!67\)\( T^{2} + \)\(64\!\cdots\!88\)\( T^{3} + \)\(86\!\cdots\!67\)\( p^{23} T^{4} + 64674463906638102342 p^{46} T^{5} + p^{69} T^{6} \)
59$S_4\times C_2$ \( 1 - 42370056643552637028 T + \)\(10\!\cdots\!13\)\( T^{2} - \)\(83\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!13\)\( p^{23} T^{4} - 42370056643552637028 p^{46} T^{5} + p^{69} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(51\!\cdots\!18\)\( T + \)\(22\!\cdots\!83\)\( T^{2} + \)\(51\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!83\)\( p^{23} T^{4} + \)\(51\!\cdots\!18\)\( p^{46} T^{5} + p^{69} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!88\)\( T + \)\(19\!\cdots\!45\)\( T^{2} - \)\(20\!\cdots\!16\)\( T^{3} + \)\(19\!\cdots\!45\)\( p^{23} T^{4} - \)\(10\!\cdots\!88\)\( p^{46} T^{5} + p^{69} T^{6} \)
71$S_4\times C_2$ \( 1 + 22690312119034642536 T + \)\(11\!\cdots\!77\)\( T^{2} + \)\(17\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!77\)\( p^{23} T^{4} + 22690312119034642536 p^{46} T^{5} + p^{69} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(41\!\cdots\!18\)\( T + \)\(26\!\cdots\!11\)\( T^{2} + \)\(61\!\cdots\!12\)\( T^{3} + \)\(26\!\cdots\!11\)\( p^{23} T^{4} + \)\(41\!\cdots\!18\)\( p^{46} T^{5} + p^{69} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(54\!\cdots\!80\)\( T + \)\(38\!\cdots\!85\)\( T^{2} + \)\(64\!\cdots\!76\)\( T^{3} + \)\(38\!\cdots\!85\)\( p^{23} T^{4} + \)\(54\!\cdots\!80\)\( p^{46} T^{5} + p^{69} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(18\!\cdots\!20\)\( T + \)\(43\!\cdots\!89\)\( T^{2} - \)\(47\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!89\)\( p^{23} T^{4} - \)\(18\!\cdots\!20\)\( p^{46} T^{5} + p^{69} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(23\!\cdots\!70\)\( T + \)\(99\!\cdots\!79\)\( T^{2} + \)\(17\!\cdots\!92\)\( T^{3} + \)\(99\!\cdots\!79\)\( p^{23} T^{4} + \)\(23\!\cdots\!70\)\( p^{46} T^{5} + p^{69} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!46\)\( T + \)\(19\!\cdots\!83\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!83\)\( p^{23} T^{4} + \)\(12\!\cdots\!46\)\( p^{46} T^{5} + p^{69} T^{6} \)
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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

−9.787722138442224381236879009476, −9.443395375831041057954334065835, −9.123891755317782052183389778009, −8.656715782074682167936258058335, −7.904712606857785606589491154585, −7.73890681429498179970622162894, −7.52285635036611255116376253613, −6.69096075312195783370549929315, −6.61563872755043413295387659608, −6.50031987077967646542957695373, −5.65508601058491431407206869060, −5.44616353877231198465113387385, −5.40502634381703444341267623531, −4.68534984962896161180718854281, −4.23501802391263894156566943377, −4.16624600318424228937705521785, −3.45980028917021805839911000869, −3.19683482033449315165030935112, −2.91163846043467559416875173345, −1.81772135894602171776111947351, −1.67042629507672000974017199352, −1.52424058356735828837256802146, −0.854986226075154746468411020697, −0.61350824281755160676593511217, −0.19022180214873994880819568927, 0.19022180214873994880819568927, 0.61350824281755160676593511217, 0.854986226075154746468411020697, 1.52424058356735828837256802146, 1.67042629507672000974017199352, 1.81772135894602171776111947351, 2.91163846043467559416875173345, 3.19683482033449315165030935112, 3.45980028917021805839911000869, 4.16624600318424228937705521785, 4.23501802391263894156566943377, 4.68534984962896161180718854281, 5.40502634381703444341267623531, 5.44616353877231198465113387385, 5.65508601058491431407206869060, 6.50031987077967646542957695373, 6.61563872755043413295387659608, 6.69096075312195783370549929315, 7.52285635036611255116376253613, 7.73890681429498179970622162894, 7.904712606857785606589491154585, 8.656715782074682167936258058335, 9.123891755317782052183389778009, 9.443395375831041057954334065835, 9.787722138442224381236879009476

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