Properties

Label 12-21e12-1.1-c1e6-0-2
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 4-s − 5·5-s + 4·6-s + 2·8-s + 10·10-s + 2·11-s + 2·12-s + 3·13-s + 10·15-s + 3·16-s − 12·17-s − 3·19-s + 5·20-s − 4·22-s − 4·24-s + 17·25-s − 6·26-s + 5·27-s − 29-s − 20·30-s + 6·31-s + 2·32-s − 4·33-s + 24·34-s + 3·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 1/2·4-s − 2.23·5-s + 1.63·6-s + 0.707·8-s + 3.16·10-s + 0.603·11-s + 0.577·12-s + 0.832·13-s + 2.58·15-s + 3/4·16-s − 2.91·17-s − 0.688·19-s + 1.11·20-s − 0.852·22-s − 0.816·24-s + 17/5·25-s − 1.17·26-s + 0.962·27-s − 0.185·29-s − 3.65·30-s + 1.07·31-s + 0.353·32-s − 0.696·33-s + 4.11·34-s + 0.493·37-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.09570975096\)
\(L(\frac12)\) \(\approx\) \(0.09570975096\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 2 T + 4 T^{2} + p T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( ( 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 + p T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 62 p T^{7} + 9 p^{2} T^{8} + 7 p^{3} T^{9} + 8 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 416 p T^{7} + 48 p^{2} T^{8} - 34 p^{3} T^{9} - 10 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 + 12 T + 54 T^{2} + 210 T^{3} + 1350 T^{4} + 5898 T^{5} + 19735 T^{6} + 5898 p T^{7} + 1350 p^{2} T^{8} + 210 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 324 p T^{7} + 468 p^{2} T^{8} - 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 758 p T^{7} + 4425 p^{2} T^{8} - 31 p^{3} T^{9} - 82 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 3 T + 69 T^{2} - 213 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 18 T + 90 T^{2} - 378 T^{3} + 7848 T^{4} - 52668 T^{5} + 160459 T^{6} - 52668 p T^{7} + 7848 p^{2} T^{8} - 378 p^{3} T^{9} + 90 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 9 T + 171 T^{2} - 999 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 162 T^{2} - 665 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 28380 p T^{7} - 3165 p^{2} T^{8} + 1209 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T - 112 T^{2} - 1238 T^{3} + 1662 T^{4} + 59806 T^{5} + 720895 T^{6} + 59806 p T^{7} + 1662 p^{2} T^{8} - 1238 p^{3} T^{9} - 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 78504 p T^{7} + 14223 p^{2} T^{8} - 573 p^{3} T^{9} - 168 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.91709597633969042317186322609, −5.79855275105721094286133080386, −5.73951986305941898485664444808, −5.55481505089618625878467967542, −5.24434896010162046004734743416, −5.19415420416481064096068118927, −4.79666396377839468859946073020, −4.61003599389768930564332478290, −4.49231979571777572635090336742, −4.46850391429830045957973243690, −4.34244755339517652840843750407, −3.78978439984548836159144949904, −3.75760701712055102898928713463, −3.65017604043491863373875568716, −3.63589354217713771305739119544, −3.30205228751117924941101815140, −2.70002845892550777932718434439, −2.63695543970514853969731333730, −2.34790073693719313483733851526, −2.27842928313334343828653154111, −1.80334589498930851154051725232, −1.13956945177663235263163715841, −0.873394054344896180135246586070, −0.53400593059500691858953867677, −0.28500900143232814639792404253, 0.28500900143232814639792404253, 0.53400593059500691858953867677, 0.873394054344896180135246586070, 1.13956945177663235263163715841, 1.80334589498930851154051725232, 2.27842928313334343828653154111, 2.34790073693719313483733851526, 2.63695543970514853969731333730, 2.70002845892550777932718434439, 3.30205228751117924941101815140, 3.63589354217713771305739119544, 3.65017604043491863373875568716, 3.75760701712055102898928713463, 3.78978439984548836159144949904, 4.34244755339517652840843750407, 4.46850391429830045957973243690, 4.49231979571777572635090336742, 4.61003599389768930564332478290, 4.79666396377839468859946073020, 5.19415420416481064096068118927, 5.24434896010162046004734743416, 5.55481505089618625878467967542, 5.73951986305941898485664444808, 5.79855275105721094286133080386, 5.91709597633969042317186322609

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

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