↑

Properties

Label 12-6e6-1.1-c20e6-0-0
Degree $12$
Conductor $46656$
Sign $1$
Analytic cond. $1.23855\times 10^{7}$
Root an. cond. $3.90010$
Motivic weight $20$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.88e4·3-s − 1.57e6·4-s − 5.66e8·7-s + 1.15e9·9-s − 2.96e10·12-s − 4.98e10·13-s + 1.64e12·16-s + 1.59e13·19-s − 1.06e13·21-s + 4.54e14·25-s − 3.20e14·27-s + 8.91e14·28-s + 1.83e15·31-s − 1.80e15·36-s + 4.24e14·37-s − 9.40e14·39-s − 4.94e16·43-s + 3.10e16·48-s − 1.71e17·49-s + 7.84e16·52-s + 3.00e17·57-s + 3.30e18·61-s − 6.51e17·63-s − 1.44e18·64-s − 2.77e18·67-s − 9.14e18·73-s + 8.57e18·75-s + ⋯
L(s)  = 1  + 0.319·3-s − 3/2·4-s − 2.00·7-s + 0.329·9-s − 0.478·12-s − 0.361·13-s + 3/2·16-s + 2.59·19-s − 0.640·21-s + 4.76·25-s − 1.55·27-s + 3.00·28-s + 2.23·31-s − 0.494·36-s + 0.0882·37-s − 0.115·39-s − 2.28·43-s + 0.478·48-s − 2.15·49-s + 0.542·52-s + 0.829·57-s + 4.62·61-s − 0.661·63-s − 5/4·64-s − 1.52·67-s − 2.12·73-s + 1.52·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+10)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(1.23855\times 10^{7}\)
Root analytic conductor: \(3.90010\)
Motivic weight: \(20\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 46656,\ (\ :[10]^{6}),\ 1)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(1.699527069\)
\(L(\frac12)\) \(\approx\) \(1.699527069\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{19} T^{2} )^{3} \)
3 \( 1 - 698 p^{3} T - 121153 p^{8} T^{2} + 24883412 p^{15} T^{3} - 121153 p^{28} T^{4} - 698 p^{43} T^{5} + p^{60} T^{6} \)
good5 \( 1 - 454857551076102 T^{2} + \)\(75\!\cdots\!11\)\( p^{3} T^{4} - \)\(73\!\cdots\!92\)\( p^{6} T^{6} + \)\(75\!\cdots\!11\)\( p^{43} T^{8} - 454857551076102 p^{80} T^{10} + p^{120} T^{12} \)
7 \( ( 1 + 40476558 p T + 29474185772602041 p T^{2} + \)\(12\!\cdots\!52\)\( p^{3} T^{3} + 29474185772602041 p^{21} T^{4} + 40476558 p^{41} T^{5} + p^{60} T^{6} )^{2} \)
11 \( 1 - \)\(28\!\cdots\!10\)\( p T^{2} + \)\(44\!\cdots\!03\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{2} T^{6} + \)\(44\!\cdots\!03\)\( p^{40} T^{8} - \)\(28\!\cdots\!10\)\( p^{81} T^{10} + p^{120} T^{12} \)
13 \( ( 1 + 24949272810 T + \)\(19\!\cdots\!31\)\( p T^{2} - \)\(88\!\cdots\!20\)\( p^{2} T^{3} + \)\(19\!\cdots\!31\)\( p^{21} T^{4} + 24949272810 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
17 \( 1 - \)\(74\!\cdots\!10\)\( T^{2} + \)\(13\!\cdots\!87\)\( p^{2} T^{4} - \)\(38\!\cdots\!20\)\( p^{4} T^{6} + \)\(13\!\cdots\!87\)\( p^{42} T^{8} - \)\(74\!\cdots\!10\)\( p^{80} T^{10} + p^{120} T^{12} \)
19 \( ( 1 - 419147297034 p T + \)\(78\!\cdots\!15\)\( p^{2} T^{2} + \)\(11\!\cdots\!40\)\( p^{3} T^{3} + \)\(78\!\cdots\!15\)\( p^{22} T^{4} - 419147297034 p^{41} T^{5} + p^{60} T^{6} )^{2} \)
23 \( 1 - \)\(66\!\cdots\!30\)\( T^{2} + \)\(20\!\cdots\!43\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{6} + \)\(20\!\cdots\!43\)\( p^{40} T^{8} - \)\(66\!\cdots\!30\)\( p^{80} T^{10} + p^{120} T^{12} \)
29 \( 1 - \)\(21\!\cdots\!66\)\( T^{2} + \)\(61\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(61\!\cdots\!55\)\( p^{40} T^{8} - \)\(21\!\cdots\!66\)\( p^{80} T^{10} + p^{120} T^{12} \)
31 \( ( 1 - 916272713198814 T + \)\(95\!\cdots\!35\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(95\!\cdots\!35\)\( p^{20} T^{4} - 916272713198814 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
37 \( ( 1 - 212166163436406 T + \)\(95\!\cdots\!11\)\( p T^{2} + \)\(42\!\cdots\!44\)\( T^{3} + \)\(95\!\cdots\!11\)\( p^{21} T^{4} - 212166163436406 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
41 \( 1 - \)\(60\!\cdots\!30\)\( T^{2} + \)\(88\!\cdots\!63\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{6} + \)\(88\!\cdots\!63\)\( p^{40} T^{8} - \)\(60\!\cdots\!30\)\( p^{80} T^{10} + p^{120} T^{12} \)
43 \( ( 1 + 24744376166975778 T + \)\(10\!\cdots\!39\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!39\)\( p^{20} T^{4} + 24744376166975778 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
47 \( 1 - \)\(94\!\cdots\!06\)\( T^{2} + \)\(42\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!15\)\( p^{40} T^{8} - \)\(94\!\cdots\!06\)\( p^{80} T^{10} + p^{120} T^{12} \)
53 \( 1 - \)\(70\!\cdots\!70\)\( T^{2} + \)\(42\!\cdots\!43\)\( T^{4} + \)\(31\!\cdots\!60\)\( T^{6} + \)\(42\!\cdots\!43\)\( p^{40} T^{8} - \)\(70\!\cdots\!70\)\( p^{80} T^{10} + p^{120} T^{12} \)
59 \( 1 - \)\(18\!\cdots\!70\)\( T^{2} + \)\(14\!\cdots\!43\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(14\!\cdots\!43\)\( p^{40} T^{8} - \)\(18\!\cdots\!70\)\( p^{80} T^{10} + p^{120} T^{12} \)
61 \( ( 1 - 1650915006436046742 T + \)\(20\!\cdots\!51\)\( T^{2} - \)\(15\!\cdots\!68\)\( T^{3} + \)\(20\!\cdots\!51\)\( p^{20} T^{4} - 1650915006436046742 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
67 \( ( 1 + 1389613136254278114 T + \)\(10\!\cdots\!27\)\( T^{2} + \)\(93\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!27\)\( p^{20} T^{4} + 1389613136254278114 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
71 \( 1 - \)\(23\!\cdots\!66\)\( T^{2} + \)\(84\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!20\)\( T^{6} + \)\(84\!\cdots\!55\)\( p^{40} T^{8} - \)\(23\!\cdots\!66\)\( p^{80} T^{10} + p^{120} T^{12} \)
73 \( ( 1 + 4574700768396250650 T + \)\(48\!\cdots\!43\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!43\)\( p^{20} T^{4} + 4574700768396250650 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
79 \( ( 1 - 45028024201628574 T + \)\(22\!\cdots\!35\)\( T^{2} + \)\(71\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!35\)\( p^{20} T^{4} - 45028024201628574 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
83 \( 1 - \)\(59\!\cdots\!62\)\( T^{2} + \)\(23\!\cdots\!11\)\( T^{4} - \)\(63\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!11\)\( p^{40} T^{8} - \)\(59\!\cdots\!62\)\( p^{80} T^{10} + p^{120} T^{12} \)
89 \( 1 - \)\(45\!\cdots\!10\)\( T^{2} + \)\(97\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!43\)\( p^{40} T^{8} - \)\(45\!\cdots\!10\)\( p^{80} T^{10} + p^{120} T^{12} \)
97 \( ( 1 - 46014452559993786918 T + \)\(10\!\cdots\!39\)\( T^{2} - \)\(32\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!39\)\( p^{20} T^{4} - 46014452559993786918 p^{40} T^{5} + p^{60} T^{6} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.822231336810565096669404867069, −8.552479299683577747152034448558, −8.355488090684715689356521127908, −7.74423773221582254248572556015, −7.40551908163622277228948329436, −6.94786738621814409515808933663, −6.93949357812049839525907219290, −6.51216389938073908373882127062, −5.94620326678040710867585477599, −5.92432762826114702856633351306, −5.06432788519136776259195733609, −4.97202153329955050031527367975, −4.76811473455732518614688099133, −4.58533525425514943290684505388, −3.61721825048883441673806316719, −3.61671645940159064098465790208, −3.19715540455368673198155432279, −2.95081155686897978026254270354, −2.91552539540165868070113004409, −2.17813237426377690151933394334, −1.54250502388213930850023374692, −1.09625541028191109385118157623, −0.925983723823246647355197967390, −0.58853202300505844344429604279, −0.21256879817809959139116301500, 0.21256879817809959139116301500, 0.58853202300505844344429604279, 0.925983723823246647355197967390, 1.09625541028191109385118157623, 1.54250502388213930850023374692, 2.17813237426377690151933394334, 2.91552539540165868070113004409, 2.95081155686897978026254270354, 3.19715540455368673198155432279, 3.61671645940159064098465790208, 3.61721825048883441673806316719, 4.58533525425514943290684505388, 4.76811473455732518614688099133, 4.97202153329955050031527367975, 5.06432788519136776259195733609, 5.92432762826114702856633351306, 5.94620326678040710867585477599, 6.51216389938073908373882127062, 6.93949357812049839525907219290, 6.94786738621814409515808933663, 7.40551908163622277228948329436, 7.74423773221582254248572556015, 8.355488090684715689356521127908, 8.552479299683577747152034448558, 8.822231336810565096669404867069

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

Plot not available for L-functions of degree greater than 10.