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Properties

Label 12-86e12-1.1-c1e6-0-0
Degree $12$
Conductor $1.637\times 10^{23}$
Sign $1$
Analytic cond. $4.24271\times 10^{10}$
Root an. cond. $7.68487$
Motivic weight $1$
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  + 2·3-s + 5·5-s − 11·9-s − 3·11-s + 5·13-s + 10·15-s − 9·17-s + 3·19-s + 3·25-s − 28·27-s + 4·29-s + 6·31-s − 6·33-s − 12·37-s + 10·39-s − 3·41-s − 55·45-s + 18·47-s − 15·49-s − 18·51-s − 4·53-s − 15·55-s + 6·57-s + 23·59-s − 10·61-s + 25·65-s + 25·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.23·5-s − 3.66·9-s − 0.904·11-s + 1.38·13-s + 2.58·15-s − 2.18·17-s + 0.688·19-s + 3/5·25-s − 5.38·27-s + 0.742·29-s + 1.07·31-s − 1.04·33-s − 1.97·37-s + 1.60·39-s − 0.468·41-s − 8.19·45-s + 2.62·47-s − 2.14·49-s − 2.52·51-s − 0.549·53-s − 2.02·55-s + 0.794·57-s + 2.99·59-s − 1.28·61-s + 3.10·65-s + 3.05·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 43^{12}\)
Sign: $1$
Analytic conductor: \(4.24271\times 10^{10}\)
Root analytic conductor: \(7.68487\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 43^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.22279464\)
\(L(\frac12)\) \(\approx\) \(11.22279464\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( ( 1 - T + 7 T^{2} - 5 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - p T + 22 T^{2} - 14 p T^{3} + 229 T^{4} - 113 p T^{5} + 1384 T^{6} - 113 p^{2} T^{7} + 229 p^{2} T^{8} - 14 p^{4} T^{9} + 22 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + 15 T^{2} - p T^{3} + 131 T^{4} - 5 p T^{5} + 986 T^{6} - 5 p^{2} T^{7} + 131 p^{2} T^{8} - p^{4} T^{9} + 15 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 + 3 T + 37 T^{2} + 102 T^{3} + 636 T^{4} + 1680 T^{5} + 7561 T^{6} + 1680 p T^{7} + 636 p^{2} T^{8} + 102 p^{3} T^{9} + 37 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 5 T + 55 T^{2} - 262 T^{3} + 114 p T^{4} - 6076 T^{5} + 24197 T^{6} - 6076 p T^{7} + 114 p^{3} T^{8} - 262 p^{3} T^{9} + 55 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 108 T^{2} + 40 p T^{3} + 4631 T^{4} + 21703 T^{5} + 104912 T^{6} + 21703 p T^{7} + 4631 p^{2} T^{8} + 40 p^{4} T^{9} + 108 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 55 T^{2} - 36 T^{3} + 1588 T^{4} - 1136 T^{5} + 39993 T^{6} - 1136 p T^{7} + 1588 p^{2} T^{8} - 36 p^{3} T^{9} + 55 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 55 T^{2} + 42 T^{3} + 2033 T^{4} + 28 p T^{5} + 55685 T^{6} + 28 p^{2} T^{7} + 2033 p^{2} T^{8} + 42 p^{3} T^{9} + 55 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 4 T + 79 T^{2} - 264 T^{3} + 3275 T^{4} - 13498 T^{5} + 111975 T^{6} - 13498 p T^{7} + 3275 p^{2} T^{8} - 264 p^{3} T^{9} + 79 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 6 T + 153 T^{2} - 800 T^{3} + 10727 T^{4} - 45604 T^{5} + 430169 T^{6} - 45604 p T^{7} + 10727 p^{2} T^{8} - 800 p^{3} T^{9} + 153 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 12 T + 241 T^{2} + 2073 T^{3} + 22853 T^{4} + 147591 T^{5} + 1132578 T^{6} + 147591 p T^{7} + 22853 p^{2} T^{8} + 2073 p^{3} T^{9} + 241 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 161 T^{2} + 496 T^{3} + 13458 T^{4} + 35964 T^{5} + 687013 T^{6} + 35964 p T^{7} + 13458 p^{2} T^{8} + 496 p^{3} T^{9} + 161 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 316 T^{2} - 3579 T^{3} + 38189 T^{4} - 313078 T^{5} + 2403635 T^{6} - 313078 p T^{7} + 38189 p^{2} T^{8} - 3579 p^{3} T^{9} + 316 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 65 T^{2} + 108 T^{3} + 2225 T^{4} - 12124 T^{5} + 113317 T^{6} - 12124 p T^{7} + 2225 p^{2} T^{8} + 108 p^{3} T^{9} + 65 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 23 T + 443 T^{2} - 5756 T^{3} + 67860 T^{4} - 631630 T^{5} + 5381071 T^{6} - 631630 p T^{7} + 67860 p^{2} T^{8} - 5756 p^{3} T^{9} + 443 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 10 T + 225 T^{2} + 1286 T^{3} + 16069 T^{4} + 43222 T^{5} + 737663 T^{6} + 43222 p T^{7} + 16069 p^{2} T^{8} + 1286 p^{3} T^{9} + 225 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 25 T + 499 T^{2} - 6898 T^{3} + 85028 T^{4} - 843010 T^{5} + 7586783 T^{6} - 843010 p T^{7} + 85028 p^{2} T^{8} - 6898 p^{3} T^{9} + 499 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 9 T + 392 T^{2} - 2660 T^{3} + 64611 T^{4} - 340299 T^{5} + 5941792 T^{6} - 340299 p T^{7} + 64611 p^{2} T^{8} - 2660 p^{3} T^{9} + 392 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 37 T + 11 p T^{2} - 12774 T^{3} + 165938 T^{4} - 1796224 T^{5} + 16604261 T^{6} - 1796224 p T^{7} + 165938 p^{2} T^{8} - 12774 p^{3} T^{9} + 11 p^{5} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T + 205 T^{2} - 210 T^{3} + 17931 T^{4} + 26761 T^{5} + 1578382 T^{6} + 26761 p T^{7} + 17931 p^{2} T^{8} - 210 p^{3} T^{9} + 205 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 4 T + 270 T^{2} - 497 T^{3} + 33183 T^{4} + 10548 T^{5} + 2914791 T^{6} + 10548 p T^{7} + 33183 p^{2} T^{8} - 497 p^{3} T^{9} + 270 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 13 T + 273 T^{2} - 928 T^{3} + 6026 T^{4} + 272808 T^{5} - 1842587 T^{6} + 272808 p T^{7} + 6026 p^{2} T^{8} - 928 p^{3} T^{9} + 273 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 51 T + 1525 T^{2} + 31574 T^{3} + 508520 T^{4} + 6591634 T^{5} + 733679 p T^{6} + 6591634 p T^{7} + 508520 p^{2} T^{8} + 31574 p^{3} T^{9} + 1525 p^{4} T^{10} + 51 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.10243988561779117921526376831, −3.64537988237080194193563027818, −3.54055422087001098536094228098, −3.48863214339301606424204026321, −3.47635863120916986086850353938, −3.38794005866123824349738924651, −3.24088990040379123533119115241, −2.91521893106317410548038633410, −2.81671902326416548324962408714, −2.79220290229231952346431945077, −2.52879726769815893900383404272, −2.50550582191261289584582730030, −2.41479088970236538930685091976, −2.23971444778461959957932690911, −2.14539903120876108628446956682, −1.86096405738752370231649046208, −1.85410350919715449928262063480, −1.69081826996819721468175961336, −1.68264159415108759694014625616, −1.28701781711546068677404590687, −0.881931476012590243763702625256, −0.61743934638455848382326439595, −0.60185446634557496454595115674, −0.55085502086389273330030604387, −0.22847697499404655061760893820, 0.22847697499404655061760893820, 0.55085502086389273330030604387, 0.60185446634557496454595115674, 0.61743934638455848382326439595, 0.881931476012590243763702625256, 1.28701781711546068677404590687, 1.68264159415108759694014625616, 1.69081826996819721468175961336, 1.85410350919715449928262063480, 1.86096405738752370231649046208, 2.14539903120876108628446956682, 2.23971444778461959957932690911, 2.41479088970236538930685091976, 2.50550582191261289584582730030, 2.52879726769815893900383404272, 2.79220290229231952346431945077, 2.81671902326416548324962408714, 2.91521893106317410548038633410, 3.24088990040379123533119115241, 3.38794005866123824349738924651, 3.47635863120916986086850353938, 3.48863214339301606424204026321, 3.54055422087001098536094228098, 3.64537988237080194193563027818, 4.10243988561779117921526376831

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

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