Properties

Label 12-1014e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.087\times 10^{18}$
Sign $1$
Analytic cond. $281767.$
Root an. cond. $2.84549$
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  + 3·2-s + 3·3-s + 3·4-s − 2·5-s + 9·6-s + 9·7-s − 2·8-s + 3·9-s − 6·10-s + 5·11-s + 9·12-s + 27·14-s − 6·15-s − 9·16-s + 8·17-s + 9·18-s − 4·19-s − 6·20-s + 27·21-s + 15·22-s − 6·24-s + 5·25-s − 2·27-s + 27·28-s − 11·29-s − 18·30-s − 10·31-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3/2·4-s − 0.894·5-s + 3.67·6-s + 3.40·7-s − 0.707·8-s + 9-s − 1.89·10-s + 1.50·11-s + 2.59·12-s + 7.21·14-s − 1.54·15-s − 9/4·16-s + 1.94·17-s + 2.12·18-s − 0.917·19-s − 1.34·20-s + 5.89·21-s + 3.19·22-s − 1.22·24-s + 25-s − 0.384·27-s + 5.10·28-s − 2.04·29-s − 3.28·30-s − 1.79·31-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(281767.\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(33.18550736\)
\(L(\frac12)\) \(\approx\) \(33.18550736\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 \)
good5 \( ( 1 + T - T^{2} - 19 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 9 T + 40 T^{2} - 17 p T^{3} + 269 T^{4} - 400 T^{5} + 575 T^{6} - 400 p T^{7} + 269 p^{2} T^{8} - 17 p^{4} T^{9} + 40 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 5 T + 97 T^{3} - 205 T^{4} - 630 T^{5} + 5083 T^{6} - 630 p T^{7} - 205 p^{2} T^{8} + 97 p^{3} T^{9} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + T^{2} + 24 T^{3} + 786 T^{4} - 1808 T^{5} - 6447 T^{6} - 1808 p T^{7} + 786 p^{2} T^{8} + 24 p^{3} T^{9} + p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T - 9 T^{2} - 4 p T^{3} - 202 T^{4} - 156 T^{5} + 3811 T^{6} - 156 p T^{7} - 202 p^{2} T^{8} - 4 p^{4} T^{9} - 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 41 T^{2} + 112 T^{3} + 738 T^{4} - 2296 T^{5} - 11193 T^{6} - 2296 p T^{7} + 738 p^{2} T^{8} + 112 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 11 T + 10 T^{2} + 3 T^{3} + 2403 T^{4} + 208 p T^{5} - 1359 p T^{6} + 208 p^{2} T^{7} + 2403 p^{2} T^{8} + 3 p^{3} T^{9} + 10 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 5 T + 43 T^{2} + 185 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 8 T - 3 T^{2} - 632 T^{3} - 2662 T^{4} + 10416 T^{5} + 188653 T^{6} + 10416 p T^{7} - 2662 p^{2} T^{8} - 632 p^{3} T^{9} - 3 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 2 T - 55 T^{2} - 254 T^{3} + 1198 T^{4} + 10326 T^{5} - 6979 T^{6} + 10326 p T^{7} + 1198 p^{2} T^{8} - 254 p^{3} T^{9} - 55 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 12 T - 5 T^{2} - 68 T^{3} + 4658 T^{4} + 6692 T^{5} - 200701 T^{6} + 6692 p T^{7} + 4658 p^{2} T^{8} - 68 p^{3} T^{9} - 5 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 4 T + 109 T^{2} + 312 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 5 T - 116 T^{2} - 141 T^{3} + 9447 T^{4} - 6142 T^{5} - 670317 T^{6} - 6142 p T^{7} + 9447 p^{2} T^{8} - 141 p^{3} T^{9} - 116 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 22 T + 149 T^{2} + 1346 T^{3} + 25526 T^{4} + 191742 T^{5} + 879417 T^{6} + 191742 p T^{7} + 25526 p^{2} T^{8} + 1346 p^{3} T^{9} + 149 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 149 T^{2} + 290 T^{3} + 17630 T^{4} - 7694 T^{5} - 1363153 T^{6} - 7694 p T^{7} + 17630 p^{2} T^{8} + 290 p^{3} T^{9} - 149 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 31 T^{2} + 178 T^{3} + 19182 T^{4} + 88394 T^{5} - 520437 T^{6} + 88394 p T^{7} + 19182 p^{2} T^{8} + 178 p^{3} T^{9} + 31 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 13 T + 189 T^{2} - 1885 T^{3} + 189 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 13 T + 121 T^{2} + 591 T^{3} + 121 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 14 T - 15 T^{2} - 1918 T^{3} - 8362 T^{4} + 99078 T^{5} + 1829149 T^{6} + 99078 p T^{7} - 8362 p^{2} T^{8} - 1918 p^{3} T^{9} - 15 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 23 T + 148 T^{2} - 355 T^{3} + 3347 T^{4} + 2612 p T^{5} + 36617 p T^{6} + 2612 p^{2} T^{7} + 3347 p^{2} T^{8} - 355 p^{3} T^{9} + 148 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.04869103132912338887535370086, −5.03915656967275562129261293691, −4.86607658766872959685883098264, −4.85354747250867772406970295180, −4.62072950083560231967966791437, −4.56566013473004950391989281375, −4.13134515670168476234517478149, −4.07071287945415083146305297097, −3.90681571887339636677644929735, −3.74007661928855701118812023176, −3.64996852849227812817929397529, −3.52713404799152067888286014426, −3.46427778106207052608353915587, −3.10522295488839398001645767557, −3.07272091389100589682155783357, −2.74839448186592878852282795610, −2.57858434348283312330631135127, −2.06493628089494876027843300420, −2.05451040647292097976538765706, −1.86821030736969631949843729010, −1.74954534537946277343199571003, −1.47265758311466854942090081515, −1.34269991152639010500229473409, −0.75554664857873964254602497006, −0.44741985832714980520524067479, 0.44741985832714980520524067479, 0.75554664857873964254602497006, 1.34269991152639010500229473409, 1.47265758311466854942090081515, 1.74954534537946277343199571003, 1.86821030736969631949843729010, 2.05451040647292097976538765706, 2.06493628089494876027843300420, 2.57858434348283312330631135127, 2.74839448186592878852282795610, 3.07272091389100589682155783357, 3.10522295488839398001645767557, 3.46427778106207052608353915587, 3.52713404799152067888286014426, 3.64996852849227812817929397529, 3.74007661928855701118812023176, 3.90681571887339636677644929735, 4.07071287945415083146305297097, 4.13134515670168476234517478149, 4.56566013473004950391989281375, 4.62072950083560231967966791437, 4.85354747250867772406970295180, 4.86607658766872959685883098264, 5.03915656967275562129261293691, 5.04869103132912338887535370086

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

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