Properties

Label 12-2151e6-1.1-c1e6-0-0
Degree $12$
Conductor $9.905\times 10^{19}$
Sign $1$
Analytic cond. $2.56746\times 10^{7}$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·4-s − 5·5-s − 9·7-s − 7·8-s − 10·10-s + 13·11-s − 13-s − 18·14-s − 3·16-s − 11·17-s − 22·19-s + 10·20-s + 26·22-s + 12·23-s − 3·25-s − 2·26-s + 18·28-s − 18·31-s + 3·32-s − 22·34-s + 45·35-s − 8·37-s − 44·38-s + 35·40-s − 10·41-s − 14·43-s + ⋯
L(s)  = 1  + 1.41·2-s − 4-s − 2.23·5-s − 3.40·7-s − 2.47·8-s − 3.16·10-s + 3.91·11-s − 0.277·13-s − 4.81·14-s − 3/4·16-s − 2.66·17-s − 5.04·19-s + 2.23·20-s + 5.54·22-s + 2.50·23-s − 3/5·25-s − 0.392·26-s + 3.40·28-s − 3.23·31-s + 0.530·32-s − 3.77·34-s + 7.60·35-s − 1.31·37-s − 7.13·38-s + 5.53·40-s − 1.56·41-s − 2.13·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 239^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 239^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 239^{6}\)
Sign: $1$
Analytic conductor: \(2.56746\times 10^{7}\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2151} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{12} \cdot 239^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( ( 1 - T )^{6} \)
good2 \( 1 - p T + 3 p T^{2} - 9 T^{3} + 19 T^{4} - 23 T^{5} + 43 T^{6} - 23 p T^{7} + 19 p^{2} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + p T + 28 T^{2} + 91 T^{3} + 311 T^{4} + 759 T^{5} + 1959 T^{6} + 759 p T^{7} + 311 p^{2} T^{8} + 91 p^{3} T^{9} + 28 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + 9 T + 59 T^{2} + 264 T^{3} + 1018 T^{4} + 3201 T^{5} + 9192 T^{6} + 3201 p T^{7} + 1018 p^{2} T^{8} + 264 p^{3} T^{9} + 59 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 13 T + 107 T^{2} - 56 p T^{3} + 2898 T^{4} - 11353 T^{5} + 40003 T^{6} - 11353 p T^{7} + 2898 p^{2} T^{8} - 56 p^{4} T^{9} + 107 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + T + 35 T^{2} + 6 T^{3} + 466 T^{4} - 41 p T^{5} + 4636 T^{6} - 41 p^{2} T^{7} + 466 p^{2} T^{8} + 6 p^{3} T^{9} + 35 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 11 T + 116 T^{2} + 759 T^{3} + 287 p T^{4} + 23469 T^{5} + 110123 T^{6} + 23469 p T^{7} + 287 p^{3} T^{8} + 759 p^{3} T^{9} + 116 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 22 T + 298 T^{2} + 2825 T^{3} + 20852 T^{4} + 122643 T^{5} + 591374 T^{6} + 122643 p T^{7} + 20852 p^{2} T^{8} + 2825 p^{3} T^{9} + 298 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 114 T^{2} - 679 T^{3} + 3436 T^{4} - 14971 T^{5} + 67362 T^{6} - 14971 p T^{7} + 3436 p^{2} T^{8} - 679 p^{3} T^{9} + 114 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 152 T^{2} - 24 T^{3} + 10099 T^{4} - 2094 T^{5} + 378855 T^{6} - 2094 p T^{7} + 10099 p^{2} T^{8} - 24 p^{3} T^{9} + 152 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 + 18 T + 249 T^{2} + 2285 T^{3} + 18296 T^{4} + 117482 T^{5} + 709487 T^{6} + 117482 p T^{7} + 18296 p^{2} T^{8} + 2285 p^{3} T^{9} + 249 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 8 T + 218 T^{2} + 1433 T^{3} + 19930 T^{4} + 104363 T^{5} + 979278 T^{6} + 104363 p T^{7} + 19930 p^{2} T^{8} + 1433 p^{3} T^{9} + 218 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 10 T + 203 T^{2} + 1712 T^{3} + 18664 T^{4} + 3108 p T^{5} + 985320 T^{6} + 3108 p^{2} T^{7} + 18664 p^{2} T^{8} + 1712 p^{3} T^{9} + 203 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 14 T + 250 T^{2} + 2009 T^{3} + 20396 T^{4} + 116083 T^{5} + 977646 T^{6} + 116083 p T^{7} + 20396 p^{2} T^{8} + 2009 p^{3} T^{9} + 250 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T + 203 T^{2} - 980 T^{3} + 14126 T^{4} - 33131 T^{5} + 636240 T^{6} - 33131 p T^{7} + 14126 p^{2} T^{8} - 980 p^{3} T^{9} + 203 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T + 258 T^{2} - 1945 T^{3} + 29750 T^{4} - 196993 T^{5} + 2001814 T^{6} - 196993 p T^{7} + 29750 p^{2} T^{8} - 1945 p^{3} T^{9} + 258 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 10 T + 279 T^{2} - 1950 T^{3} + 32290 T^{4} - 2900 p T^{5} + 2280580 T^{6} - 2900 p^{2} T^{7} + 32290 p^{2} T^{8} - 1950 p^{3} T^{9} + 279 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 347 T^{2} + 3219 T^{3} + 50608 T^{4} + 367892 T^{5} + 4048483 T^{6} + 367892 p T^{7} + 50608 p^{2} T^{8} + 3219 p^{3} T^{9} + 347 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 36 T + 837 T^{2} + 13760 T^{3} + 180475 T^{4} + 1921076 T^{5} + 17209166 T^{6} + 1921076 p T^{7} + 180475 p^{2} T^{8} + 13760 p^{3} T^{9} + 837 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T + 143 T^{2} - 162 T^{3} + 14694 T^{4} - 28195 T^{5} + 1333060 T^{6} - 28195 p T^{7} + 14694 p^{2} T^{8} - 162 p^{3} T^{9} + 143 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 32 T + 740 T^{2} + 12129 T^{3} + 164176 T^{4} + 1799525 T^{5} + 16869046 T^{6} + 1799525 p T^{7} + 164176 p^{2} T^{8} + 12129 p^{3} T^{9} + 740 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + T + 140 T^{2} + 543 T^{3} + 14128 T^{4} + 16468 T^{5} + 1433338 T^{6} + 16468 p T^{7} + 14128 p^{2} T^{8} + 543 p^{3} T^{9} + 140 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 7 T + 85 T^{2} + 186 T^{3} + 7186 T^{4} - 62435 T^{5} + 1306061 T^{6} - 62435 p T^{7} + 7186 p^{2} T^{8} + 186 p^{3} T^{9} + 85 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 17 T + 360 T^{2} + 5159 T^{3} + 67070 T^{4} + 722148 T^{5} + 7651022 T^{6} + 722148 p T^{7} + 67070 p^{2} T^{8} + 5159 p^{3} T^{9} + 360 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 28 T + 583 T^{2} + 8216 T^{3} + 108992 T^{4} + 1225176 T^{5} + 13191524 T^{6} + 1225176 p T^{7} + 108992 p^{2} T^{8} + 8216 p^{3} T^{9} + 583 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.11008524350250157022591583044, −4.65054950404136265284566193363, −4.59862996213378650155705500281, −4.55495908899498687617438815417, −4.48839934083428248930867094401, −4.44015031778319853542313118208, −4.19500101962515673629936425440, −4.16727604521443044754487572500, −3.90097865914353830232270829547, −3.87347592667321329959606672035, −3.78119295870544768717971258865, −3.68589512581740716016483647849, −3.52591331309711736655606884272, −3.30760451202736388203106044519, −3.23825607727718463093019462989, −2.96991051792693047461880823104, −2.95968876355688432363099676661, −2.60301233676385610031142503513, −2.25711911582769317092470742776, −2.20051754880824364210673340836, −2.00705383684722046822785918950, −1.56666171496230014263661095266, −1.52783842239567128542582705179, −1.51044915052966091444512786167, −1.05092594542681997969987050282, 0, 0, 0, 0, 0, 0, 1.05092594542681997969987050282, 1.51044915052966091444512786167, 1.52783842239567128542582705179, 1.56666171496230014263661095266, 2.00705383684722046822785918950, 2.20051754880824364210673340836, 2.25711911582769317092470742776, 2.60301233676385610031142503513, 2.95968876355688432363099676661, 2.96991051792693047461880823104, 3.23825607727718463093019462989, 3.30760451202736388203106044519, 3.52591331309711736655606884272, 3.68589512581740716016483647849, 3.78119295870544768717971258865, 3.87347592667321329959606672035, 3.90097865914353830232270829547, 4.16727604521443044754487572500, 4.19500101962515673629936425440, 4.44015031778319853542313118208, 4.48839934083428248930867094401, 4.55495908899498687617438815417, 4.59862996213378650155705500281, 4.65054950404136265284566193363, 5.11008524350250157022591583044

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

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