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Properties

Label 12-4005e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.127\times 10^{21}$
Sign $1$
Analytic cond. $1.06973\times 10^{9}$
Root an. cond. $5.65509$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·4-s + 6·5-s + 7-s + 3·8-s + 24·10-s − 3·13-s + 4·14-s − 3·16-s + 13·17-s + 12·19-s + 36·20-s + 19·23-s + 21·25-s − 12·26-s + 6·28-s + 10·31-s − 7·32-s + 52·34-s + 6·35-s − 37-s + 48·38-s + 18·40-s + 4·41-s − 7·43-s + 76·46-s + 15·47-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s + 2.68·5-s + 0.377·7-s + 1.06·8-s + 7.58·10-s − 0.832·13-s + 1.06·14-s − 3/4·16-s + 3.15·17-s + 2.75·19-s + 8.04·20-s + 3.96·23-s + 21/5·25-s − 2.35·26-s + 1.13·28-s + 1.79·31-s − 1.23·32-s + 8.91·34-s + 1.01·35-s − 0.164·37-s + 7.78·38-s + 2.84·40-s + 0.624·41-s − 1.06·43-s + 11.2·46-s + 2.18·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 89^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 89^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 89^{6}\)
Sign: $1$
Analytic conductor: \(1.06973\times 10^{9}\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 89^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(308.7000264\)
\(L(\frac12)\) \(\approx\) \(308.7000264\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - T )^{6} \)
89 \( ( 1 - T )^{6} \)
good2 \( 1 - p^{2} T + 5 p T^{2} - 19 T^{3} + 31 T^{4} - 45 T^{5} + p^{6} T^{6} - 45 p T^{7} + 31 p^{2} T^{8} - 19 p^{3} T^{9} + 5 p^{5} T^{10} - p^{7} T^{11} + p^{6} T^{12} \)
7 \( 1 - T + 22 T^{2} - 4 p T^{3} + 271 T^{4} - 359 T^{5} + 2196 T^{6} - 359 p T^{7} + 271 p^{2} T^{8} - 4 p^{4} T^{9} + 22 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 + p T^{2} )^{6} \)
13 \( 1 + 3 T + 58 T^{2} + 178 T^{3} + 1565 T^{4} + 4439 T^{5} + 25448 T^{6} + 4439 p T^{7} + 1565 p^{2} T^{8} + 178 p^{3} T^{9} + 58 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 13 T + 124 T^{2} - 842 T^{3} + 4779 T^{4} - 23057 T^{5} + 100512 T^{6} - 23057 p T^{7} + 4779 p^{2} T^{8} - 842 p^{3} T^{9} + 124 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 - 2 T + p T^{2} )^{6} \)
23 \( 1 - 19 T + 252 T^{2} - 2321 T^{3} + 17551 T^{4} - 107206 T^{5} + 563016 T^{6} - 107206 p T^{7} + 17551 p^{2} T^{8} - 2321 p^{3} T^{9} + 252 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 119 T^{2} - 157 T^{3} + 6406 T^{4} - 12779 T^{5} + 220820 T^{6} - 12779 p T^{7} + 6406 p^{2} T^{8} - 157 p^{3} T^{9} + 119 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 - 10 T + 146 T^{2} - 1046 T^{3} + 9487 T^{4} - 53004 T^{5} + 363260 T^{6} - 53004 p T^{7} + 9487 p^{2} T^{8} - 1046 p^{3} T^{9} + 146 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + T + 174 T^{2} + 106 T^{3} + 13685 T^{4} + 5017 T^{5} + 637552 T^{6} + 5017 p T^{7} + 13685 p^{2} T^{8} + 106 p^{3} T^{9} + 174 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 129 T^{2} - 797 T^{3} + 8170 T^{4} - 62491 T^{5} + 369120 T^{6} - 62491 p T^{7} + 8170 p^{2} T^{8} - 797 p^{3} T^{9} + 129 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 7 T + 186 T^{2} + 1240 T^{3} + 16187 T^{4} + 97741 T^{5} + 864292 T^{6} + 97741 p T^{7} + 16187 p^{2} T^{8} + 1240 p^{3} T^{9} + 186 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 308 T^{2} - 3202 T^{3} + 37463 T^{4} - 287727 T^{5} + 2368936 T^{6} - 287727 p T^{7} + 37463 p^{2} T^{8} - 3202 p^{3} T^{9} + 308 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 27 T + 492 T^{2} - 6228 T^{3} + 65801 T^{4} - 571977 T^{5} + 4474228 T^{6} - 571977 p T^{7} + 65801 p^{2} T^{8} - 6228 p^{3} T^{9} + 492 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T + 315 T^{2} - 1033 T^{3} + 43290 T^{4} - 114053 T^{5} + 3325944 T^{6} - 114053 p T^{7} + 43290 p^{2} T^{8} - 1033 p^{3} T^{9} + 315 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T + 274 T^{2} - 2296 T^{3} + 34455 T^{4} - 272864 T^{5} + 2617340 T^{6} - 272864 p T^{7} + 34455 p^{2} T^{8} - 2296 p^{3} T^{9} + 274 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 11 T + 198 T^{2} + 2001 T^{3} + 22799 T^{4} + 207466 T^{5} + 1932836 T^{6} + 207466 p T^{7} + 22799 p^{2} T^{8} + 2001 p^{3} T^{9} + 198 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 16 T + 358 T^{2} - 3656 T^{3} + 49359 T^{4} - 382232 T^{5} + 4137460 T^{6} - 382232 p T^{7} + 49359 p^{2} T^{8} - 3656 p^{3} T^{9} + 358 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - T - 4 T^{2} - 153 T^{3} + 10007 T^{4} - 25070 T^{5} - 61448 T^{6} - 25070 p T^{7} + 10007 p^{2} T^{8} - 153 p^{3} T^{9} - 4 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 353 T^{2} + 39 T^{3} + 59178 T^{4} + 6667 T^{5} + 5920008 T^{6} + 6667 p T^{7} + 59178 p^{2} T^{8} + 39 p^{3} T^{9} + 353 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 17 T + 550 T^{2} - 6923 T^{3} + 120111 T^{4} - 1138518 T^{5} + 13504612 T^{6} - 1138518 p T^{7} + 120111 p^{2} T^{8} - 6923 p^{3} T^{9} + 550 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 29 T + 716 T^{2} + 12281 T^{3} + 185895 T^{4} + 2253258 T^{5} + 24385176 T^{6} + 2253258 p T^{7} + 185895 p^{2} T^{8} + 12281 p^{3} T^{9} + 716 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.61744974597789883669535663863, −4.17540197329886521415262795370, −3.93586335065835651904223789833, −3.89244341340169613292240533072, −3.88886611672412415993770973966, −3.79662616514103077687082426501, −3.45401377505032646725156415551, −3.19244340920515616590076105107, −3.17017345940920468303402613457, −3.09337926844969837190067680003, −2.96644241517708619011662626087, −2.88207887628285819460904448663, −2.79667221025084835965051893401, −2.46912647108449284381044597622, −2.37307539256546262589194062558, −2.18158146186237671657229330522, −2.08200325769082879246528953886, −1.63409165616849202095504975802, −1.57924656397591606048666994316, −1.35373891129565240411406520826, −1.32052206705900042066118636466, −0.879947830219348062939190558424, −0.78583423156555059230673448507, −0.71698288404846958156307271099, −0.69490746642242851662255696656, 0.69490746642242851662255696656, 0.71698288404846958156307271099, 0.78583423156555059230673448507, 0.879947830219348062939190558424, 1.32052206705900042066118636466, 1.35373891129565240411406520826, 1.57924656397591606048666994316, 1.63409165616849202095504975802, 2.08200325769082879246528953886, 2.18158146186237671657229330522, 2.37307539256546262589194062558, 2.46912647108449284381044597622, 2.79667221025084835965051893401, 2.88207887628285819460904448663, 2.96644241517708619011662626087, 3.09337926844969837190067680003, 3.17017345940920468303402613457, 3.19244340920515616590076105107, 3.45401377505032646725156415551, 3.79662616514103077687082426501, 3.88886611672412415993770973966, 3.89244341340169613292240533072, 3.93586335065835651904223789833, 4.17540197329886521415262795370, 4.61744974597789883669535663863

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

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