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Properties

Label 12-3808e6-1.1-c1e6-0-8
Degree $12$
Conductor $3.049\times 10^{21}$
Sign $1$
Analytic cond. $7.90395\times 10^{8}$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 6·5-s − 6·7-s − 5·9-s + 2·11-s − 4·13-s − 12·15-s − 6·17-s + 10·19-s − 12·21-s − 4·23-s + 5·25-s − 12·27-s − 14·29-s − 8·31-s + 4·33-s + 36·35-s − 4·37-s − 8·39-s − 2·41-s + 8·43-s + 30·45-s + 2·47-s + 21·49-s − 12·51-s − 10·53-s − 12·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.68·5-s − 2.26·7-s − 5/3·9-s + 0.603·11-s − 1.10·13-s − 3.09·15-s − 1.45·17-s + 2.29·19-s − 2.61·21-s − 0.834·23-s + 25-s − 2.30·27-s − 2.59·29-s − 1.43·31-s + 0.696·33-s + 6.08·35-s − 0.657·37-s − 1.28·39-s − 0.312·41-s + 1.21·43-s + 4.47·45-s + 0.291·47-s + 3·49-s − 1.68·51-s − 1.37·53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(7.90395\times 10^{8}\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( 1 \)
7 \( ( 1 + T )^{6} \)
17 \( ( 1 + T )^{6} \)
good3 \( 1 - 2 T + p^{2} T^{2} - 16 T^{3} + 16 p T^{4} - 68 T^{5} + 56 p T^{6} - 68 p T^{7} + 16 p^{3} T^{8} - 16 p^{3} T^{9} + p^{6} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 31 T^{2} + 122 T^{3} + 16 p^{2} T^{4} + 1106 T^{5} + 2688 T^{6} + 1106 p T^{7} + 16 p^{4} T^{8} + 122 p^{3} T^{9} + 31 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T + 32 T^{2} - 18 T^{3} + 515 T^{4} + 40 T^{5} + 6232 T^{6} + 40 p T^{7} + 515 p^{2} T^{8} - 18 p^{3} T^{9} + 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 4 p T^{2} + 116 T^{3} + 971 T^{4} + 80 p T^{5} + 12048 T^{6} + 80 p^{2} T^{7} + 971 p^{2} T^{8} + 116 p^{3} T^{9} + 4 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 10 T + 82 T^{2} - 326 T^{3} + 1027 T^{4} + 852 T^{5} - 6444 T^{6} + 852 p T^{7} + 1027 p^{2} T^{8} - 326 p^{3} T^{9} + 82 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T + 2 p T^{2} + 148 T^{3} + 1163 T^{4} + 2656 T^{5} + 23980 T^{6} + 2656 p T^{7} + 1163 p^{2} T^{8} + 148 p^{3} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 14 T + 220 T^{2} + 1906 T^{3} + 17179 T^{4} + 106472 T^{5} + 675888 T^{6} + 106472 p T^{7} + 17179 p^{2} T^{8} + 1906 p^{3} T^{9} + 220 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 89 T^{2} + 664 T^{3} + 4656 T^{4} + 25704 T^{5} + 169068 T^{6} + 25704 p T^{7} + 4656 p^{2} T^{8} + 664 p^{3} T^{9} + 89 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 4 T + 122 T^{2} + 500 T^{3} + 8227 T^{4} + 28312 T^{5} + 365892 T^{6} + 28312 p T^{7} + 8227 p^{2} T^{8} + 500 p^{3} T^{9} + 122 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T + 195 T^{2} + 182 T^{3} + 16804 T^{4} + 6552 T^{5} + 861264 T^{6} + 6552 p T^{7} + 16804 p^{2} T^{8} + 182 p^{3} T^{9} + 195 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 187 T^{2} - 1114 T^{3} + 16156 T^{4} - 78342 T^{5} + 862296 T^{6} - 78342 p T^{7} + 16156 p^{2} T^{8} - 1114 p^{3} T^{9} + 187 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 2 T + 192 T^{2} - 250 T^{3} + 17923 T^{4} - 17448 T^{5} + 1041080 T^{6} - 17448 p T^{7} + 17923 p^{2} T^{8} - 250 p^{3} T^{9} + 192 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 10 T + 225 T^{2} + 1956 T^{3} + 23880 T^{4} + 174814 T^{5} + 1566748 T^{6} + 174814 p T^{7} + 23880 p^{2} T^{8} + 1956 p^{3} T^{9} + 225 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 14 T + 214 T^{2} - 2370 T^{3} + 25991 T^{4} - 223436 T^{5} + 1918964 T^{6} - 223436 p T^{7} + 25991 p^{2} T^{8} - 2370 p^{3} T^{9} + 214 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 20 T + 263 T^{2} + 1898 T^{3} + 3984 T^{4} - 82348 T^{5} - 1046808 T^{6} - 82348 p T^{7} + 3984 p^{2} T^{8} + 1898 p^{3} T^{9} + 263 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T + 271 T^{2} - 1610 T^{3} + 35028 T^{4} - 188628 T^{5} + 2861664 T^{6} - 188628 p T^{7} + 35028 p^{2} T^{8} - 1610 p^{3} T^{9} + 271 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 20 T + 446 T^{2} + 5884 T^{3} + 79003 T^{4} + 765512 T^{5} + 7422124 T^{6} + 765512 p T^{7} + 79003 p^{2} T^{8} + 5884 p^{3} T^{9} + 446 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T + 347 T^{2} + 1634 T^{3} + 53720 T^{4} + 200976 T^{5} + 4919912 T^{6} + 200976 p T^{7} + 53720 p^{2} T^{8} + 1634 p^{3} T^{9} + 347 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 38 T + 740 T^{2} + 9566 T^{3} + 97155 T^{4} + 876392 T^{5} + 7764912 T^{6} + 876392 p T^{7} + 97155 p^{2} T^{8} + 9566 p^{3} T^{9} + 740 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 16 T + 376 T^{2} - 5088 T^{3} + 66491 T^{4} - 748248 T^{5} + 7035736 T^{6} - 748248 p T^{7} + 66491 p^{2} T^{8} - 5088 p^{3} T^{9} + 376 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T + 196 T^{2} - 1210 T^{3} + 25163 T^{4} - 107880 T^{5} + 3544352 T^{6} - 107880 p T^{7} + 25163 p^{2} T^{8} - 1210 p^{3} T^{9} + 196 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 12 T + 435 T^{2} + 4682 T^{3} + 89128 T^{4} + 812778 T^{5} + 10912024 T^{6} + 812778 p T^{7} + 89128 p^{2} T^{8} + 4682 p^{3} T^{9} + 435 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.68988468230978140532815495230, −4.48085928532999538008344230528, −4.26164874240943837375038611753, −4.18198948711529641343310973155, −4.14916377796797257388094119950, −4.03955590068256630080634915285, −3.95744246604668837048185629908, −3.76890133138156239608886851705, −3.55275701668609376636345551378, −3.51311772738782730522167282922, −3.38222716255013513359990443942, −3.32156312193299551930375717380, −3.11700656696090303244193741452, −2.92667837783300233222455776100, −2.72074148856322114960099326989, −2.62368430784210112605618436452, −2.57942132105659418605102994758, −2.43064576003601370561302715364, −2.35403189183865894639525150762, −1.88087861536073928105515110741, −1.75881643655682183272083048447, −1.62998570287081830709253050734, −1.28650857694416440978908914032, −1.07078851395833386115171509513, −1.03480855013030031661941402520, 0, 0, 0, 0, 0, 0, 1.03480855013030031661941402520, 1.07078851395833386115171509513, 1.28650857694416440978908914032, 1.62998570287081830709253050734, 1.75881643655682183272083048447, 1.88087861536073928105515110741, 2.35403189183865894639525150762, 2.43064576003601370561302715364, 2.57942132105659418605102994758, 2.62368430784210112605618436452, 2.72074148856322114960099326989, 2.92667837783300233222455776100, 3.11700656696090303244193741452, 3.32156312193299551930375717380, 3.38222716255013513359990443942, 3.51311772738782730522167282922, 3.55275701668609376636345551378, 3.76890133138156239608886851705, 3.95744246604668837048185629908, 4.03955590068256630080634915285, 4.14916377796797257388094119950, 4.18198948711529641343310973155, 4.26164874240943837375038611753, 4.48085928532999538008344230528, 4.68988468230978140532815495230

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

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