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Properties

Label 12-882e6-1.1-c1e6-0-9
Degree $12$
Conductor $4.708\times 10^{17}$
Sign $1$
Analytic cond. $122032.$
Root an. cond. $2.65382$
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  − 3·2-s + 2·3-s + 3·4-s + 5·5-s − 6·6-s + 2·8-s + 6·9-s − 15·10-s − 11-s + 6·12-s + 2·13-s + 10·15-s − 9·16-s − 8·17-s − 18·18-s − 6·19-s + 15·20-s + 3·22-s − 7·23-s + 4·24-s + 19·25-s − 6·26-s + 7·27-s − 5·29-s − 30·30-s + 14·31-s + 9·32-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 3/2·4-s + 2.23·5-s − 2.44·6-s + 0.707·8-s + 2·9-s − 4.74·10-s − 0.301·11-s + 1.73·12-s + 0.554·13-s + 2.58·15-s − 9/4·16-s − 1.94·17-s − 4.24·18-s − 1.37·19-s + 3.35·20-s + 0.639·22-s − 1.45·23-s + 0.816·24-s + 19/5·25-s − 1.17·26-s + 1.34·27-s − 0.928·29-s − 5.47·30-s + 2.51·31-s + 1.59·32-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(122032.\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.346326616\)
\(L(\frac12)\) \(\approx\) \(4.346326616\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 - 2 T - 2 T^{2} + p^{2} T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good5 \( 1 - p T + 6 T^{2} - T^{3} + 31 T^{4} - 68 T^{5} + 29 T^{6} - 68 p T^{7} + 31 p^{2} T^{8} - p^{3} T^{9} + 6 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 32 p T^{5} + 457 p T^{6} + 32 p^{2} T^{7} - 83 p^{2} T^{8} - 103 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 2 T - 32 T^{2} + 2 p T^{3} + 730 T^{4} - 230 T^{5} - 10729 T^{6} - 230 p T^{7} + 730 p^{2} T^{8} + 2 p^{4} T^{9} - 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 4 T + 7 T^{2} - 32 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 3484 p T^{7} + 1417 p^{2} T^{8} - 127 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 6020 p T^{7} - 185 p^{2} T^{8} - 371 p^{3} T^{9} - 30 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 58 T^{2} - 250 T^{3} + 2992 T^{4} - 9728 T^{5} - 11857 T^{6} - 9728 p T^{7} + 2992 p^{2} T^{8} - 250 p^{3} T^{9} + 58 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 9 T + 102 T^{2} - 593 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 12 T - 18 T^{2} + 78 T^{3} + 7470 T^{4} - 24546 T^{5} - 158105 T^{6} - 24546 p T^{7} + 7470 p^{2} T^{8} + 78 p^{3} T^{9} - 18 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T - 108 T^{2} - 267 T^{3} + 7263 T^{4} + 9786 T^{5} - 360137 T^{6} + 9786 p T^{7} + 7263 p^{2} T^{8} - 267 p^{3} T^{9} - 108 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 9 T + 117 T^{2} + 963 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 4 T - 60 T^{2} - 994 T^{3} - 1304 T^{4} + 464 p T^{5} + 7381 p T^{6} + 464 p^{2} T^{7} - 1304 p^{2} T^{8} - 994 p^{3} T^{9} - 60 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 32 T^{2} + 650 T^{3} + 292 T^{4} - 19532 T^{5} + 306323 T^{6} - 19532 p T^{7} + 292 p^{2} T^{8} + 650 p^{3} T^{9} - 32 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 25 T + 371 T^{2} + 3601 T^{3} + 371 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 32420 p T^{7} + 29404 p^{2} T^{8} - 518 p^{3} T^{9} - 180 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 9 T + 261 T^{2} + 1539 T^{3} + 261 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 28 T + 257 T^{2} - 2820 T^{3} + 59506 T^{4} - 545924 T^{5} + 3126001 T^{6} - 545924 p T^{7} + 59506 p^{2} T^{8} - 2820 p^{3} T^{9} + 257 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.46900018579840163620851965525, −5.31781205151041342751254757703, −5.02084635469026355603953107960, −4.62853911398916652674298496364, −4.57515989992684649891162188458, −4.48107074577826830496612455077, −4.29106550015204442582309994485, −4.28962694290251497999541177025, −4.20865674559227692349734677090, −4.20788426783118535793088224913, −3.57146123119803564193335167612, −3.35931554086640352305274923772, −3.09647240771912127959814468508, −2.85472356020979899481084827048, −2.77989156945957558011047296340, −2.76572925270284119181190736177, −2.24212920294888074247423571400, −2.02134690490939877612695741801, −1.94983388478908225270867261004, −1.90669855114734031407795588638, −1.73456223095055306752323702489, −1.34867612637518316615419232847, −0.823122546367163808942760802401, −0.790007284465544302429867852647, −0.54296962872212431414735931939, 0.54296962872212431414735931939, 0.790007284465544302429867852647, 0.823122546367163808942760802401, 1.34867612637518316615419232847, 1.73456223095055306752323702489, 1.90669855114734031407795588638, 1.94983388478908225270867261004, 2.02134690490939877612695741801, 2.24212920294888074247423571400, 2.76572925270284119181190736177, 2.77989156945957558011047296340, 2.85472356020979899481084827048, 3.09647240771912127959814468508, 3.35931554086640352305274923772, 3.57146123119803564193335167612, 4.20788426783118535793088224913, 4.20865674559227692349734677090, 4.28962694290251497999541177025, 4.29106550015204442582309994485, 4.48107074577826830496612455077, 4.57515989992684649891162188458, 4.62853911398916652674298496364, 5.02084635469026355603953107960, 5.31781205151041342751254757703, 5.46900018579840163620851965525

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

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