Properties

Label 12-9196e6-1.1-c1e6-0-1
Degree $12$
Conductor $6.048\times 10^{23}$
Sign $1$
Analytic cond. $1.56767\times 10^{11}$
Root an. cond. $8.56915$
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  + 3-s + 5·5-s + 2·7-s − 8·13-s + 5·15-s + 2·17-s + 6·19-s + 2·21-s + 5·23-s + 4·25-s + 27-s − 10·29-s + 3·31-s + 10·35-s + 7·37-s − 8·39-s − 6·41-s − 16·43-s − 13·49-s + 2·51-s + 20·53-s + 6·57-s + 15·59-s − 24·61-s − 40·65-s + 25·67-s + 5·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.23·5-s + 0.755·7-s − 2.21·13-s + 1.29·15-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 1.04·23-s + 4/5·25-s + 0.192·27-s − 1.85·29-s + 0.538·31-s + 1.69·35-s + 1.15·37-s − 1.28·39-s − 0.937·41-s − 2.43·43-s − 1.85·49-s + 0.280·51-s + 2.74·53-s + 0.794·57-s + 1.95·59-s − 3.07·61-s − 4.96·65-s + 3.05·67-s + 0.601·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 11^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.56767\times 10^{11}\)
Root analytic conductor: \(8.56915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 11^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24.73123629\)
\(L(\frac12)\) \(\approx\) \(24.73123629\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
19 \( ( 1 - T )^{6} \)
good3 \( 1 - T + T^{2} - 2 T^{3} + 2 p T^{5} + 2 p T^{6} + 2 p^{2} T^{7} - 2 p^{3} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - p T + 21 T^{2} - 48 T^{3} + 96 T^{4} - 86 T^{5} + 178 T^{6} - 86 p T^{7} + 96 p^{2} T^{8} - 48 p^{3} T^{9} + 21 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 17 T^{2} - 12 T^{3} + 116 T^{4} + 82 T^{5} + 548 T^{6} + 82 p T^{7} + 116 p^{2} T^{8} - 12 p^{3} T^{9} + 17 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 8 T + 55 T^{2} + 274 T^{3} + 1148 T^{4} + 4180 T^{5} + 15612 T^{6} + 4180 p T^{7} + 1148 p^{2} T^{8} + 274 p^{3} T^{9} + 55 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T + 30 T^{2} - 122 T^{3} + 607 T^{4} - 2084 T^{5} + 12868 T^{6} - 2084 p T^{7} + 607 p^{2} T^{8} - 122 p^{3} T^{9} + 30 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 5 T + 50 T^{2} - 243 T^{3} + 1967 T^{4} - 8550 T^{5} + 54684 T^{6} - 8550 p T^{7} + 1967 p^{2} T^{8} - 243 p^{3} T^{9} + 50 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 10 T + 83 T^{2} + 422 T^{3} + 2572 T^{4} + 8554 T^{5} + 46924 T^{6} + 8554 p T^{7} + 2572 p^{2} T^{8} + 422 p^{3} T^{9} + 83 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T + 107 T^{2} - 78 T^{3} + 4418 T^{4} + 6464 T^{5} + 127702 T^{6} + 6464 p T^{7} + 4418 p^{2} T^{8} - 78 p^{3} T^{9} + 107 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 7 T + 112 T^{2} - 967 T^{3} + 8191 T^{4} - 54478 T^{5} + 394928 T^{6} - 54478 p T^{7} + 8191 p^{2} T^{8} - 967 p^{3} T^{9} + 112 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 203 T^{2} + 766 T^{3} + 16684 T^{4} + 41824 T^{5} + 822548 T^{6} + 41824 p T^{7} + 16684 p^{2} T^{8} + 766 p^{3} T^{9} + 203 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 16 T + 209 T^{2} + 1924 T^{3} + 18072 T^{4} + 134610 T^{5} + 980204 T^{6} + 134610 p T^{7} + 18072 p^{2} T^{8} + 1924 p^{3} T^{9} + 209 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 158 T^{2} + 176 T^{3} + 13039 T^{4} + 13104 T^{5} + 746020 T^{6} + 13104 p T^{7} + 13039 p^{2} T^{8} + 176 p^{3} T^{9} + 158 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 20 T + 370 T^{2} - 4436 T^{3} + 49815 T^{4} - 434728 T^{5} + 3532444 T^{6} - 434728 p T^{7} + 49815 p^{2} T^{8} - 4436 p^{3} T^{9} + 370 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 15 T + 248 T^{2} - 3029 T^{3} + 31879 T^{4} - 292034 T^{5} + 2417840 T^{6} - 292034 p T^{7} + 31879 p^{2} T^{8} - 3029 p^{3} T^{9} + 248 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 24 T + 426 T^{2} + 5624 T^{3} + 63015 T^{4} + 601104 T^{5} + 4996652 T^{6} + 601104 p T^{7} + 63015 p^{2} T^{8} + 5624 p^{3} T^{9} + 426 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 25 T + 479 T^{2} - 5946 T^{3} + 65282 T^{4} - 570676 T^{5} + 4992830 T^{6} - 570676 p T^{7} + 65282 p^{2} T^{8} - 5946 p^{3} T^{9} + 479 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T + 269 T^{2} + 1862 T^{3} + 35908 T^{4} + 199430 T^{5} + 3020546 T^{6} + 199430 p T^{7} + 35908 p^{2} T^{8} + 1862 p^{3} T^{9} + 269 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 26 T + 458 T^{2} - 6474 T^{3} + 1031 p T^{4} - 760652 T^{5} + 6945740 T^{6} - 760652 p T^{7} + 1031 p^{3} T^{8} - 6474 p^{3} T^{9} + 458 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 16 T + 454 T^{2} - 5264 T^{3} + 82511 T^{4} - 740480 T^{5} + 8351892 T^{6} - 740480 p T^{7} + 82511 p^{2} T^{8} - 5264 p^{3} T^{9} + 454 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 2 T + 421 T^{2} - 744 T^{3} + 79316 T^{4} - 117434 T^{5} + 102428 p T^{6} - 117434 p T^{7} + 79316 p^{2} T^{8} - 744 p^{3} T^{9} + 421 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 7 T + 308 T^{2} - 1931 T^{3} + 43111 T^{4} - 259078 T^{5} + 4205512 T^{6} - 259078 p T^{7} + 43111 p^{2} T^{8} - 1931 p^{3} T^{9} + 308 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 37 T + 916 T^{2} - 16581 T^{3} + 246255 T^{4} - 3080646 T^{5} + 32767192 T^{6} - 3080646 p T^{7} + 246255 p^{2} T^{8} - 16581 p^{3} T^{9} + 916 p^{4} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.74880922899040425221968544352, −3.74301728588056231539003333160, −3.59737750447278896843902853793, −3.58918191433433462474299929352, −3.41002590540326031575575348435, −3.24361335283491094268522599549, −3.06546868282280441265336217317, −2.96492202204678729056406587700, −2.81683814901502365136588976498, −2.57492360677729809891743874544, −2.50075095351376076360500111574, −2.31011876214535916601158543362, −2.27060655434224588389586979573, −2.05353532433346521566657175669, −2.04676837653148180190298256077, −1.95295852371940405162471581926, −1.68204206098905578555238127514, −1.61679478065239104608485466535, −1.40910753242949966180163605166, −1.29756604352299163656107365630, −0.917487193278197447093505296668, −0.77351046281811203432085666999, −0.73530211297840959185255807652, −0.33265589163497194137293581398, −0.32832648709508235527474515308, 0.32832648709508235527474515308, 0.33265589163497194137293581398, 0.73530211297840959185255807652, 0.77351046281811203432085666999, 0.917487193278197447093505296668, 1.29756604352299163656107365630, 1.40910753242949966180163605166, 1.61679478065239104608485466535, 1.68204206098905578555238127514, 1.95295852371940405162471581926, 2.04676837653148180190298256077, 2.05353532433346521566657175669, 2.27060655434224588389586979573, 2.31011876214535916601158543362, 2.50075095351376076360500111574, 2.57492360677729809891743874544, 2.81683814901502365136588976498, 2.96492202204678729056406587700, 3.06546868282280441265336217317, 3.24361335283491094268522599549, 3.41002590540326031575575348435, 3.58918191433433462474299929352, 3.59737750447278896843902853793, 3.74301728588056231539003333160, 3.74880922899040425221968544352

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

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