Properties

Label 12-21e12-1.1-c1e6-0-8
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

\(L(1)\) \(\approx\) \(2.718310108\)
\(L(\frac12)\) \(\approx\) \(2.718310108\)
\(L(\frac{3}{2})\) 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

−6.08691504350663686541410982176, −5.86251386362576078051762755759, −5.60566595737931378758026433268, −5.43017540144501146058442154720, −5.42032082552388186942971863224, −5.23659024610299446090665615220, −5.11097080909143445566771258165, −4.72888735364825148093065206085, −4.50142330143137551517289606004, −4.45355292663987074359726640677, −4.21143269761757994171133612556, −3.88442579329640054648194177024, −3.63849217342501234715875153699, −3.37425689220625176328824464038, −3.12636563099194181617887947327, −3.08045787666799646134911828145, −2.92582215667111395665157219090, −2.83732570488504292474643333126, −2.10451806765471578826005338199, −2.09913181216847841192600381992, −1.92044022032468868539042285289, −1.76992940563860351208040119292, −0.948980920962569836788598051200, −0.891407145659966308056892614784, −0.67161604404873578446540322970, 0.67161604404873578446540322970, 0.891407145659966308056892614784, 0.948980920962569836788598051200, 1.76992940563860351208040119292, 1.92044022032468868539042285289, 2.09913181216847841192600381992, 2.10451806765471578826005338199, 2.83732570488504292474643333126, 2.92582215667111395665157219090, 3.08045787666799646134911828145, 3.12636563099194181617887947327, 3.37425689220625176328824464038, 3.63849217342501234715875153699, 3.88442579329640054648194177024, 4.21143269761757994171133612556, 4.45355292663987074359726640677, 4.50142330143137551517289606004, 4.72888735364825148093065206085, 5.11097080909143445566771258165, 5.23659024610299446090665615220, 5.42032082552388186942971863224, 5.43017540144501146058442154720, 5.60566595737931378758026433268, 5.86251386362576078051762755759, 6.08691504350663686541410982176

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

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