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Properties

Label 12-7098e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.279\times 10^{23}$
Sign $1$
Analytic cond. $3.31496\times 10^{10}$
Root an. cond. $7.52846$
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  − 6·2-s − 6·3-s + 21·4-s + 3·5-s + 36·6-s − 6·7-s − 56·8-s + 21·9-s − 18·10-s + 6·11-s − 126·12-s + 36·14-s − 18·15-s + 126·16-s + 10·17-s − 126·18-s − 8·19-s + 63·20-s + 36·21-s − 36·22-s − 12·23-s + 336·24-s − 8·25-s − 56·27-s − 126·28-s + 4·29-s + 108·30-s + ⋯
L(s)  = 1  − 4.24·2-s − 3.46·3-s + 21/2·4-s + 1.34·5-s + 14.6·6-s − 2.26·7-s − 19.7·8-s + 7·9-s − 5.69·10-s + 1.80·11-s − 36.3·12-s + 9.62·14-s − 4.64·15-s + 63/2·16-s + 2.42·17-s − 29.6·18-s − 1.83·19-s + 14.0·20-s + 7.85·21-s − 7.67·22-s − 2.50·23-s + 68.5·24-s − 8/5·25-s − 10.7·27-s − 23.8·28-s + 0.742·29-s + 19.7·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(3.31496\times 10^{10}\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
3 \( ( 1 + T )^{6} \)
7 \( ( 1 + T )^{6} \)
13 \( 1 \)
good5 \( 1 - 3 T + 17 T^{2} - 44 T^{3} + 154 T^{4} - 68 p T^{5} + 899 T^{6} - 68 p^{2} T^{7} + 154 p^{2} T^{8} - 44 p^{3} T^{9} + 17 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 31 T^{2} - 150 T^{3} + 751 T^{4} - 2648 T^{5} + 8825 T^{6} - 2648 p T^{7} + 751 p^{2} T^{8} - 150 p^{3} T^{9} + 31 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 10 T + 89 T^{2} - 560 T^{3} + 3399 T^{4} - 16272 T^{5} + 74329 T^{6} - 16272 p T^{7} + 3399 p^{2} T^{8} - 560 p^{3} T^{9} + 89 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 8 T + 65 T^{2} + 324 T^{3} + 1705 T^{4} + 6396 T^{5} + 30205 T^{6} + 6396 p T^{7} + 1705 p^{2} T^{8} + 324 p^{3} T^{9} + 65 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 155 T^{2} + 1140 T^{3} + 8411 T^{4} + 45238 T^{5} + 246367 T^{6} + 45238 p T^{7} + 8411 p^{2} T^{8} + 1140 p^{3} T^{9} + 155 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 63 T^{2} - 317 T^{3} + 3551 T^{4} - 13511 T^{5} + 112994 T^{6} - 13511 p T^{7} + 3551 p^{2} T^{8} - 317 p^{3} T^{9} + 63 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 7 T + 81 T^{2} + 224 T^{3} + 268 T^{4} - 364 p T^{5} - 78847 T^{6} - 364 p^{2} T^{7} + 268 p^{2} T^{8} + 224 p^{3} T^{9} + 81 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T + 55 T^{2} + 118 T^{3} + 2228 T^{4} + 4826 T^{5} + 115337 T^{6} + 4826 p T^{7} + 2228 p^{2} T^{8} + 118 p^{3} T^{9} + 55 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 97 T^{2} + 18 T^{3} + 5890 T^{4} + 350 T^{5} + 292859 T^{6} + 350 p T^{7} + 5890 p^{2} T^{8} + 18 p^{3} T^{9} + 97 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 174 T^{2} - 4 T^{3} + 11565 T^{4} - 30019 T^{5} + 509584 T^{6} - 30019 p T^{7} + 11565 p^{2} T^{8} - 4 p^{3} T^{9} + 174 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 29 T + 535 T^{2} + 7138 T^{3} + 76437 T^{4} + 673729 T^{5} + 5028318 T^{6} + 673729 p T^{7} + 76437 p^{2} T^{8} + 7138 p^{3} T^{9} + 535 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 133 T^{2} + 771 T^{3} + 481 T^{4} - 38759 T^{5} - 395422 T^{6} - 38759 p T^{7} + 481 p^{2} T^{8} + 771 p^{3} T^{9} + 133 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T + 321 T^{2} + 467 T^{3} + 44415 T^{4} + 48229 T^{5} + 3417270 T^{6} + 48229 p T^{7} + 44415 p^{2} T^{8} + 467 p^{3} T^{9} + 321 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 13 T + 219 T^{2} - 1608 T^{3} + 19317 T^{4} - 125407 T^{5} + 1383214 T^{6} - 125407 p T^{7} + 19317 p^{2} T^{8} - 1608 p^{3} T^{9} + 219 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 22 T + 7 p T^{2} + 6071 T^{3} + 77809 T^{4} + 734269 T^{5} + 6874266 T^{6} + 734269 p T^{7} + 77809 p^{2} T^{8} + 6071 p^{3} T^{9} + 7 p^{5} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + T + 182 T^{2} - 666 T^{3} + 15525 T^{4} - 122451 T^{5} + 1119280 T^{6} - 122451 p T^{7} + 15525 p^{2} T^{8} - 666 p^{3} T^{9} + 182 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 29 T + 522 T^{2} + 6922 T^{3} + 79991 T^{4} + 11241 p T^{5} + 7543396 T^{6} + 11241 p^{2} T^{7} + 79991 p^{2} T^{8} + 6922 p^{3} T^{9} + 522 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 619 T^{2} + 9331 T^{3} + 137203 T^{4} + 1461931 T^{5} + 14938458 T^{6} + 1461931 p T^{7} + 137203 p^{2} T^{8} + 9331 p^{3} T^{9} + 619 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 7 T + 270 T^{2} + 1386 T^{3} + 36497 T^{4} + 147007 T^{5} + 3497032 T^{6} + 147007 p T^{7} + 36497 p^{2} T^{8} + 1386 p^{3} T^{9} + 270 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 11 T + 387 T^{2} - 3318 T^{3} + 69458 T^{4} - 486872 T^{5} + 7659001 T^{6} - 486872 p T^{7} + 69458 p^{2} T^{8} - 3318 p^{3} T^{9} + 387 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 155 T^{2} - 587 T^{3} + 33363 T^{4} - 84169 T^{5} + 2796906 T^{6} - 84169 p T^{7} + 33363 p^{2} T^{8} - 587 p^{3} T^{9} + 155 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.50176609153761543287707722283, −4.10138406795038327563114837505, −4.09366979716204584477657808103, −3.96599193816035784291236901432, −3.83537531288032581822658396698, −3.82521940642726538430730076535, −3.79062379681426999150777309500, −3.19719320234788997351758836032, −3.14435507761957628733508401326, −3.14265901421268765568479133967, −3.10151415124789702506962088089, −2.99081627461552607396073961333, −2.83848979248380154864606527743, −2.20527047672282686035984061226, −2.15510280092794019607211905560, −2.05484978066290853076561361474, −2.02638050549750800616513902467, −1.90085968206951949521185575291, −1.78519732346117088631556033540, −1.45717367898828177798531691910, −1.38858528889119188516771734820, −1.25118304985493210006643994589, −1.03688270272818813268388824536, −0.999536137210637947788048340515, −0.844979153868344675175342802477, 0, 0, 0, 0, 0, 0, 0.844979153868344675175342802477, 0.999536137210637947788048340515, 1.03688270272818813268388824536, 1.25118304985493210006643994589, 1.38858528889119188516771734820, 1.45717367898828177798531691910, 1.78519732346117088631556033540, 1.90085968206951949521185575291, 2.02638050549750800616513902467, 2.05484978066290853076561361474, 2.15510280092794019607211905560, 2.20527047672282686035984061226, 2.83848979248380154864606527743, 2.99081627461552607396073961333, 3.10151415124789702506962088089, 3.14265901421268765568479133967, 3.14435507761957628733508401326, 3.19719320234788997351758836032, 3.79062379681426999150777309500, 3.82521940642726538430730076535, 3.83537531288032581822658396698, 3.96599193816035784291236901432, 4.09366979716204584477657808103, 4.10138406795038327563114837505, 4.50176609153761543287707722283

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

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