Properties

Label 12-405e6-1.1-c3e6-0-1
Degree $12$
Conductor $4.413\times 10^{15}$
Sign $1$
Analytic cond. $1.86177\times 10^{8}$
Root an. cond. $4.88833$
Motivic weight $3$
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  − 5·2-s + 16·4-s − 15·5-s + 4·7-s − 15·8-s + 75·10-s − 5·11-s − 7·13-s − 20·14-s − 47·16-s + 310·17-s − 100·19-s − 240·20-s + 25·22-s − 285·23-s + 75·25-s + 35·26-s + 64·28-s − 115·29-s + 115·31-s + 240·32-s − 1.55e3·34-s − 60·35-s − 768·37-s + 500·38-s + 225·40-s − 580·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 2·4-s − 1.34·5-s + 0.215·7-s − 0.662·8-s + 2.37·10-s − 0.137·11-s − 0.149·13-s − 0.381·14-s − 0.734·16-s + 4.42·17-s − 1.20·19-s − 2.68·20-s + 0.242·22-s − 2.58·23-s + 3/5·25-s + 0.264·26-s + 0.431·28-s − 0.736·29-s + 0.666·31-s + 1.32·32-s − 7.81·34-s − 0.289·35-s − 3.41·37-s + 2.13·38-s + 0.889·40-s − 2.20·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.86177\times 10^{8}\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 5^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.07140254605\)
\(L(\frac12)\) \(\approx\) \(0.07140254605\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + p T + p^{2} T^{2} )^{3} \)
good2 \( 1 + 5 T + 9 T^{2} - 5 p^{2} T^{3} - 61 p T^{4} - 5 p^{5} T^{5} - 31 p^{2} T^{6} - 5 p^{8} T^{7} - 61 p^{7} T^{8} - 5 p^{11} T^{9} + 9 p^{12} T^{10} + 5 p^{15} T^{11} + p^{18} T^{12} \)
7 \( 1 - 4 T - 218 T^{2} - 12044 T^{3} - 470 p T^{4} + 1471136 T^{5} + 90890438 T^{6} + 1471136 p^{3} T^{7} - 470 p^{7} T^{8} - 12044 p^{9} T^{9} - 218 p^{12} T^{10} - 4 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 + 5 T - 1080 T^{2} + 41425 T^{3} - 159980 T^{4} - 25684495 T^{5} + 3368624582 T^{6} - 25684495 p^{3} T^{7} - 159980 p^{6} T^{8} + 41425 p^{9} T^{9} - 1080 p^{12} T^{10} + 5 p^{15} T^{11} + p^{18} T^{12} \)
13 \( 1 + 7 T - 5773 T^{2} - 33612 T^{3} + 20844469 T^{4} + 71194013 T^{5} - 52113757154 T^{6} + 71194013 p^{3} T^{7} + 20844469 p^{6} T^{8} - 33612 p^{9} T^{9} - 5773 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12} \)
17 \( ( 1 - 155 T + 20347 T^{2} - 1564790 T^{3} + 20347 p^{3} T^{4} - 155 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
19 \( ( 1 + 50 T + 206 p T^{2} + 317888 T^{3} + 206 p^{4} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
23 \( 1 + 285 T + 28524 T^{2} + 2256405 T^{3} + 317400684 T^{4} + 13020224865 T^{5} - 1488512348966 T^{6} + 13020224865 p^{3} T^{7} + 317400684 p^{6} T^{8} + 2256405 p^{9} T^{9} + 28524 p^{12} T^{10} + 285 p^{15} T^{11} + p^{18} T^{12} \)
29 \( 1 + 115 T - 12534 T^{2} + 4624025 T^{3} + 4511720 p T^{4} - 97492036145 T^{5} + 5655360898904 T^{6} - 97492036145 p^{3} T^{7} + 4511720 p^{7} T^{8} + 4624025 p^{9} T^{9} - 12534 p^{12} T^{10} + 115 p^{15} T^{11} + p^{18} T^{12} \)
31 \( 1 - 115 T - 46916 T^{2} + 4911037 T^{3} + 1145212360 T^{4} - 45632382451 T^{5} - 31639275386878 T^{6} - 45632382451 p^{3} T^{7} + 1145212360 p^{6} T^{8} + 4911037 p^{9} T^{9} - 46916 p^{12} T^{10} - 115 p^{15} T^{11} + p^{18} T^{12} \)
37 \( ( 1 + 384 T + 84036 T^{2} + 16234306 T^{3} + 84036 p^{3} T^{4} + 384 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( 1 + 580 T + 39825 T^{2} + 4282220 T^{3} + 18874413850 T^{4} + 3916827775780 T^{5} + 28208543319497 T^{6} + 3916827775780 p^{3} T^{7} + 18874413850 p^{6} T^{8} + 4282220 p^{9} T^{9} + 39825 p^{12} T^{10} + 580 p^{15} T^{11} + p^{18} T^{12} \)
43 \( 1 - 797 T + 254168 T^{2} - 60937233 T^{3} + 18159967960 T^{4} - 2996533058677 T^{5} + 178186071529462 T^{6} - 2996533058677 p^{3} T^{7} + 18159967960 p^{6} T^{8} - 60937233 p^{9} T^{9} + 254168 p^{12} T^{10} - 797 p^{15} T^{11} + p^{18} T^{12} \)
47 \( 1 - 145 T - 41292 T^{2} + 79957855 T^{3} - 9038581352 T^{4} - 2459618174785 T^{5} + 3588415823907962 T^{6} - 2459618174785 p^{3} T^{7} - 9038581352 p^{6} T^{8} + 79957855 p^{9} T^{9} - 41292 p^{12} T^{10} - 145 p^{15} T^{11} + p^{18} T^{12} \)
53 \( ( 1 + 400 T + 388459 T^{2} + 106443280 T^{3} + 388459 p^{3} T^{4} + 400 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
59 \( 1 + 380 T - 444345 T^{2} - 78042740 T^{3} + 168369306070 T^{4} + 12964114409180 T^{5} - 37187829488683597 T^{6} + 12964114409180 p^{3} T^{7} + 168369306070 p^{6} T^{8} - 78042740 p^{9} T^{9} - 444345 p^{12} T^{10} + 380 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 - 152 T - 570364 T^{2} + 37659180 T^{3} + 207780466900 T^{4} - 4822768441984 T^{5} - 53658262585089842 T^{6} - 4822768441984 p^{3} T^{7} + 207780466900 p^{6} T^{8} + 37659180 p^{9} T^{9} - 570364 p^{12} T^{10} - 152 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 + 2 T - 658286 T^{2} + 40449076 T^{3} + 235395581326 T^{4} - 13491305732662 T^{5} - 75538814997629578 T^{6} - 13491305732662 p^{3} T^{7} + 235395581326 p^{6} T^{8} + 40449076 p^{9} T^{9} - 658286 p^{12} T^{10} + 2 p^{15} T^{11} + p^{18} T^{12} \)
71 \( ( 1 - 40 T + 396361 T^{2} + 187438400 T^{3} + 396361 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
73 \( ( 1 + 980 T + 1431584 T^{2} + 778921274 T^{3} + 1431584 p^{3} T^{4} + 980 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
79 \( 1 + 1013 T - 505163 T^{2} - 265360862 T^{3} + 669863306119 T^{4} + 107513622088673 T^{5} - 340902024884391226 T^{6} + 107513622088673 p^{3} T^{7} + 669863306119 p^{6} T^{8} - 265360862 p^{9} T^{9} - 505163 p^{12} T^{10} + 1013 p^{15} T^{11} + p^{18} T^{12} \)
83 \( 1 + 270 T - 1341033 T^{2} - 66689730 T^{3} + 1130197048818 T^{4} - 31258807306650 T^{5} - 742796527633986737 T^{6} - 31258807306650 p^{3} T^{7} + 1130197048818 p^{6} T^{8} - 66689730 p^{9} T^{9} - 1341033 p^{12} T^{10} + 270 p^{15} T^{11} + p^{18} T^{12} \)
89 \( ( 1 - 1020 T + 2161707 T^{2} - 1313072760 T^{3} + 2161707 p^{3} T^{4} - 1020 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
97 \( 1 + 720 T - 2185056 T^{2} - 627148804 T^{3} + 3914793681408 T^{4} + 525582536940432 T^{5} - 3925878298041416250 T^{6} + 525582536940432 p^{3} T^{7} + 3914793681408 p^{6} T^{8} - 627148804 p^{9} T^{9} - 2185056 p^{12} T^{10} + 720 p^{15} T^{11} + p^{18} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.62547380080525757078993569934, −5.45561204288180215502115786392, −5.38795242752034981059773314225, −5.14784276610183533018549465027, −4.92480281325066858510860014159, −4.65507733153298550982040049496, −4.48291851025284081658966766969, −4.34034338598726265666252481410, −3.94112664548772255025201716174, −3.86097591586383647691639689788, −3.64824043422593475547455410892, −3.60283005806627806457985529169, −3.31529663861330577177185364737, −3.05705475293480062842552897615, −2.98138116172490221368962435327, −2.42568079306610098152788029925, −2.25073774080615322608996484684, −2.15980097699267147441947046555, −1.69230472040377580775777739907, −1.50303581040574459981454260874, −1.25637683492598498581851342842, −1.18969607481503369864633214036, −0.887283689555789082636028240071, −0.14377683630341645679550903822, −0.13642427571285517820669134909, 0.13642427571285517820669134909, 0.14377683630341645679550903822, 0.887283689555789082636028240071, 1.18969607481503369864633214036, 1.25637683492598498581851342842, 1.50303581040574459981454260874, 1.69230472040377580775777739907, 2.15980097699267147441947046555, 2.25073774080615322608996484684, 2.42568079306610098152788029925, 2.98138116172490221368962435327, 3.05705475293480062842552897615, 3.31529663861330577177185364737, 3.60283005806627806457985529169, 3.64824043422593475547455410892, 3.86097591586383647691639689788, 3.94112664548772255025201716174, 4.34034338598726265666252481410, 4.48291851025284081658966766969, 4.65507733153298550982040049496, 4.92480281325066858510860014159, 5.14784276610183533018549465027, 5.38795242752034981059773314225, 5.45561204288180215502115786392, 5.62547380080525757078993569934

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

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