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Properties

Label 12-1372e6-1.1-c1e6-0-1
Degree $12$
Conductor $6.670\times 10^{18}$
Sign $1$
Analytic cond. $1.72896\times 10^{6}$
Root an. cond. $3.30990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s + 7·5-s + 20·9-s + 3·11-s + 7·13-s + 49·15-s + 7·19-s + 11·23-s + 11·25-s + 28·27-s + 6·29-s + 14·31-s + 21·33-s − 6·37-s + 49·39-s − 3·43-s + 140·45-s + 42·47-s − 17·53-s + 21·55-s + 49·57-s + 35·59-s + 7·61-s + 49·65-s − 14·67-s + 77·69-s − 28·71-s + ⋯
L(s)  = 1  + 4.04·3-s + 3.13·5-s + 20/3·9-s + 0.904·11-s + 1.94·13-s + 12.6·15-s + 1.60·19-s + 2.29·23-s + 11/5·25-s + 5.38·27-s + 1.11·29-s + 2.51·31-s + 3.65·33-s − 0.986·37-s + 7.84·39-s − 0.457·43-s + 20.8·45-s + 6.12·47-s − 2.33·53-s + 2.83·55-s + 6.49·57-s + 4.55·59-s + 0.896·61-s + 6.07·65-s − 1.71·67-s + 9.26·69-s − 3.32·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 7^{18}\)
Sign: $1$
Analytic conductor: \(1.72896\times 10^{6}\)
Root analytic conductor: \(3.30990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 7^{18} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(139.0279633\)
\(L(\frac12)\) \(\approx\) \(139.0279633\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 7 T + 29 T^{2} - 91 T^{3} + 235 T^{4} - 511 T^{5} + 955 T^{6} - 511 p T^{7} + 235 p^{2} T^{8} - 91 p^{3} T^{9} + 29 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 7 T + 38 T^{2} - 147 T^{3} + 491 T^{4} - 1358 T^{5} + 3301 T^{6} - 1358 p T^{7} + 491 p^{2} T^{8} - 147 p^{3} T^{9} + 38 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 26 T^{2} - 129 T^{3} + 514 T^{4} - 2111 T^{5} + 7245 T^{6} - 2111 p T^{7} + 514 p^{2} T^{8} - 129 p^{3} T^{9} + 26 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 7 T + 66 T^{2} - 329 T^{3} + 1882 T^{4} - 7357 T^{5} + 31367 T^{6} - 7357 p T^{7} + 1882 p^{2} T^{8} - 329 p^{3} T^{9} + 66 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 p T^{2} - 77 T^{3} + 2133 T^{4} - 3465 T^{5} + 43393 T^{6} - 3465 p T^{7} + 2133 p^{2} T^{8} - 77 p^{3} T^{9} + 4 p^{5} T^{10} + p^{6} T^{12} \)
19 \( 1 - 7 T + 4 p T^{2} - 455 T^{3} + 2859 T^{4} - 14798 T^{5} + 67907 T^{6} - 14798 p T^{7} + 2859 p^{2} T^{8} - 455 p^{3} T^{9} + 4 p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 11 T + 126 T^{2} - 923 T^{3} + 6254 T^{4} - 35385 T^{5} + 180307 T^{6} - 35385 p T^{7} + 6254 p^{2} T^{8} - 923 p^{3} T^{9} + 126 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 140 T^{2} - 652 T^{3} + 8707 T^{4} - 32403 T^{5} + 318655 T^{6} - 32403 p T^{7} + 8707 p^{2} T^{8} - 652 p^{3} T^{9} + 140 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 144 T^{2} - 1092 T^{3} + 8423 T^{4} - 54719 T^{5} + 336547 T^{6} - 54719 p T^{7} + 8423 p^{2} T^{8} - 1092 p^{3} T^{9} + 144 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 6 T + 104 T^{2} + 640 T^{3} + 7118 T^{4} + 36234 T^{5} + 311885 T^{6} + 36234 p T^{7} + 7118 p^{2} T^{8} + 640 p^{3} T^{9} + 104 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 110 T^{2} - 371 T^{3} + 5175 T^{4} - 40677 T^{5} + 194485 T^{6} - 40677 p T^{7} + 5175 p^{2} T^{8} - 371 p^{3} T^{9} + 110 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 T + 78 T^{2} + 623 T^{3} + 5354 T^{4} + 28123 T^{5} + 338645 T^{6} + 28123 p T^{7} + 5354 p^{2} T^{8} + 623 p^{3} T^{9} + 78 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 42 T + 977 T^{2} - 15596 T^{3} + 188196 T^{4} - 1785315 T^{5} + 13619413 T^{6} - 1785315 p T^{7} + 188196 p^{2} T^{8} - 15596 p^{3} T^{9} + 977 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 17 T + 355 T^{2} + 3699 T^{3} + 44138 T^{4} + 335192 T^{5} + 2970625 T^{6} + 335192 p T^{7} + 44138 p^{2} T^{8} + 3699 p^{3} T^{9} + 355 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 35 T + 762 T^{2} - 11648 T^{3} + 142748 T^{4} - 1422645 T^{5} + 11969815 T^{6} - 1422645 p T^{7} + 142748 p^{2} T^{8} - 11648 p^{3} T^{9} + 762 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 7 T + 216 T^{2} - 1337 T^{3} + 26960 T^{4} - 136689 T^{5} + 1991353 T^{6} - 136689 p T^{7} + 26960 p^{2} T^{8} - 1337 p^{3} T^{9} + 216 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T + 304 T^{2} + 3220 T^{3} + 42198 T^{4} + 350434 T^{5} + 3547277 T^{6} + 350434 p T^{7} + 42198 p^{2} T^{8} + 3220 p^{3} T^{9} + 304 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 14 T + 262 T^{2} + 1995 T^{3} + 262 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 169 T^{2} - 35 T^{3} + 19457 T^{4} - 11746 T^{5} + 1678925 T^{6} - 11746 p T^{7} + 19457 p^{2} T^{8} - 35 p^{3} T^{9} + 169 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 6 T + 349 T^{2} - 1431 T^{3} + 52093 T^{4} - 150852 T^{5} + 4861045 T^{6} - 150852 p T^{7} + 52093 p^{2} T^{8} - 1431 p^{3} T^{9} + 349 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 49 T + 1410 T^{2} - 28364 T^{3} + 439683 T^{4} - 5431356 T^{5} + 54724299 T^{6} - 5431356 p T^{7} + 439683 p^{2} T^{8} - 28364 p^{3} T^{9} + 1410 p^{4} T^{10} - 49 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 7 T + 132 T^{2} - 182 T^{3} + 13324 T^{4} + 37219 T^{5} + 2029965 T^{6} + 37219 p T^{7} + 13324 p^{2} T^{8} - 182 p^{3} T^{9} + 132 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 21 T + 569 T^{2} + 7791 T^{3} + 128179 T^{4} + 1346709 T^{5} + 16281947 T^{6} + 1346709 p T^{7} + 128179 p^{2} T^{8} + 7791 p^{3} T^{9} + 569 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.19289845826566826057970195702, −4.95038413641049450153035987850, −4.47656088001398442080636330144, −4.46716163583017543950114407506, −4.32235680491958520798069941369, −4.17003458801688465039610137084, −4.14606132903698217173299365922, −3.56802085735891056604047382189, −3.49595699679349981109315962718, −3.43863490874569508096703956793, −3.34967274463033990216276989418, −3.34211434293892248847665267468, −3.03570132955106407482430656157, −2.65507935084041857380888463194, −2.61234849957967884562388944585, −2.45031710587531048922602275419, −2.41061636457466312771764054386, −2.25521342655386501754462234847, −2.22332975164725986356542963353, −1.72980781101519418138824536609, −1.55524932485917444470736761180, −1.34677997827451176845653195870, −1.05844669913017489339669193991, −0.930569531880088682581424118698, −0.808306601652134000730420664322, 0.808306601652134000730420664322, 0.930569531880088682581424118698, 1.05844669913017489339669193991, 1.34677997827451176845653195870, 1.55524932485917444470736761180, 1.72980781101519418138824536609, 2.22332975164725986356542963353, 2.25521342655386501754462234847, 2.41061636457466312771764054386, 2.45031710587531048922602275419, 2.61234849957967884562388944585, 2.65507935084041857380888463194, 3.03570132955106407482430656157, 3.34211434293892248847665267468, 3.34967274463033990216276989418, 3.43863490874569508096703956793, 3.49595699679349981109315962718, 3.56802085735891056604047382189, 4.14606132903698217173299365922, 4.17003458801688465039610137084, 4.32235680491958520798069941369, 4.46716163583017543950114407506, 4.47656088001398442080636330144, 4.95038413641049450153035987850, 5.19289845826566826057970195702

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

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