Properties

Label 12-5292e6-1.1-c1e6-0-7
Degree $12$
Conductor $2.196\times 10^{22}$
Sign $1$
Analytic cond. $5.69353\times 10^{9}$
Root an. cond. $6.50052$
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  − 2·5-s − 4·11-s − 3·13-s + 2·17-s − 3·19-s − 28·23-s − 19·25-s + 29-s + 3·31-s + 3·37-s − 3·43-s + 21·47-s + 6·53-s + 8·55-s + 31·59-s − 6·61-s + 6·65-s − 6·67-s − 34·71-s + 3·73-s + 9·79-s + 20·83-s − 4·85-s + 12·89-s + 6·95-s + 9·97-s − 26·101-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s − 0.832·13-s + 0.485·17-s − 0.688·19-s − 5.83·23-s − 3.79·25-s + 0.185·29-s + 0.538·31-s + 0.493·37-s − 0.457·43-s + 3.06·47-s + 0.824·53-s + 1.07·55-s + 4.03·59-s − 0.768·61-s + 0.744·65-s − 0.733·67-s − 4.03·71-s + 0.351·73-s + 1.01·79-s + 2.19·83-s − 0.433·85-s + 1.27·89-s + 0.615·95-s + 0.913·97-s − 2.58·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(5.69353\times 10^{9}\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.109386715\)
\(L(\frac12)\) \(\approx\) \(1.109386715\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + T + 11 T^{2} + 9 T^{3} + 11 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 2 T + 8 T^{2} - 15 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 3 T + 3 T^{2} - 84 T^{3} - 15 p T^{4} + 345 T^{5} + 5006 T^{6} + 345 p T^{7} - 15 p^{3} T^{8} - 84 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T - 28 T^{2} - 22 T^{3} + 438 T^{4} + 926 T^{5} - 8297 T^{6} + 926 p T^{7} + 438 p^{2} T^{8} - 22 p^{3} T^{9} - 28 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 24 T^{2} + 29 T^{3} + 357 T^{4} - 1524 T^{5} - 8997 T^{6} - 1524 p T^{7} + 357 p^{2} T^{8} + 29 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 14 T + 122 T^{2} + 675 T^{3} + 122 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - T - 46 T^{2} - 149 T^{3} + 897 T^{4} + 4282 T^{5} - 13523 T^{6} + 4282 p T^{7} + 897 p^{2} T^{8} - 149 p^{3} T^{9} - 46 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 1344 p T^{7} + 1005 p^{2} T^{8} + 147 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 72 T^{2} + 155 T^{3} + 2967 T^{4} - 2244 T^{5} - 114171 T^{6} - 2244 p T^{7} + 2967 p^{2} T^{8} + 155 p^{3} T^{9} - 72 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 14820 p T^{7} - 1947 p^{2} T^{8} - 979 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 21 T + 180 T^{2} - 1119 T^{3} + 10053 T^{4} - 100416 T^{5} + 788551 T^{6} - 100416 p T^{7} + 10053 p^{2} T^{8} - 1119 p^{3} T^{9} + 180 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 126 T^{2} + 282 T^{3} + 13896 T^{4} - 15396 T^{5} - 801173 T^{6} - 15396 p T^{7} + 13896 p^{2} T^{8} + 282 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 31 T + 476 T^{2} - 5741 T^{3} + 62553 T^{4} - 587576 T^{5} + 4781851 T^{6} - 587576 p T^{7} + 62553 p^{2} T^{8} - 5741 p^{3} T^{9} + 476 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 35394 p T^{7} + 3018 p^{2} T^{8} + 642 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 28236 p T^{7} + 17268 p^{2} T^{8} - 506 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 17 T + 119 T^{2} + 507 T^{3} + 119 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 186 T^{2} + 133 T^{3} + 22713 T^{4} - 582 T^{5} - 1916871 T^{6} - 582 p T^{7} + 22713 p^{2} T^{8} + 133 p^{3} T^{9} - 186 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 15786 p T^{7} + 13143 p^{2} T^{8} + 351 p^{3} T^{9} - 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 20 T + 38 T^{2} - 346 T^{3} + 32058 T^{4} - 183754 T^{5} - 606869 T^{6} - 183754 p T^{7} + 32058 p^{2} T^{8} - 346 p^{3} T^{9} + 38 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 12 T - 72 T^{2} + 258 T^{3} + 10332 T^{4} + 58524 T^{5} - 1852445 T^{6} + 58524 p T^{7} + 10332 p^{2} T^{8} + 258 p^{3} T^{9} - 72 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 73950 p T^{7} - 7707 p^{2} T^{8} + 2023 p^{3} T^{9} - 66 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.09992400235426595471585753710, −3.98777975653233517822357665810, −3.82733899648690135246109846404, −3.81814045795120704091102484720, −3.79892733960165986659890562620, −3.68482171503282083742080513819, −3.48433706320733597387637128022, −3.32962794296751094530872644662, −2.90710366775551102757480493694, −2.87603217428234682590172662074, −2.58898803852175604265998908977, −2.58120877366627035255740001180, −2.55652352747526705254068207282, −2.26514394622200435146606102671, −2.09646070867582257704586287824, −1.97660204511248777004726491455, −1.89492014085407569796128799838, −1.84655799272422411743994243777, −1.49558502680031541556242891620, −1.38628258311960928711102103834, −1.05004526977893453861449988297, −0.54680698122942634311996271494, −0.49739671117448717280588803998, −0.40060167921662194071853160009, −0.18628751706720502297019825778, 0.18628751706720502297019825778, 0.40060167921662194071853160009, 0.49739671117448717280588803998, 0.54680698122942634311996271494, 1.05004526977893453861449988297, 1.38628258311960928711102103834, 1.49558502680031541556242891620, 1.84655799272422411743994243777, 1.89492014085407569796128799838, 1.97660204511248777004726491455, 2.09646070867582257704586287824, 2.26514394622200435146606102671, 2.55652352747526705254068207282, 2.58120877366627035255740001180, 2.58898803852175604265998908977, 2.87603217428234682590172662074, 2.90710366775551102757480493694, 3.32962794296751094530872644662, 3.48433706320733597387637128022, 3.68482171503282083742080513819, 3.79892733960165986659890562620, 3.81814045795120704091102484720, 3.82733899648690135246109846404, 3.98777975653233517822357665810, 4.09992400235426595471585753710

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

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