↑

Properties

Label 12-1875e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.345\times 10^{19}$
Sign $1$
Analytic cond. $1.12634\times 10^{7}$
Root an. cond. $3.86936$
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-s − 6·3-s − 6·6-s − 2·7-s + 8-s + 21·9-s − 2·14-s + 2·17-s + 21·18-s − 2·19-s + 12·21-s − 23-s − 6·24-s − 56·27-s + 31·29-s − 2·31-s + 3·32-s + 2·34-s − 22·37-s − 2·38-s + 33·41-s + 12·42-s − 3·43-s − 46-s − 6·47-s − 17·49-s − 12·51-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s − 2.44·6-s − 0.755·7-s + 0.353·8-s + 7·9-s − 0.534·14-s + 0.485·17-s + 4.94·18-s − 0.458·19-s + 2.61·21-s − 0.208·23-s − 1.22·24-s − 10.7·27-s + 5.75·29-s − 0.359·31-s + 0.530·32-s + 0.342·34-s − 3.61·37-s − 0.324·38-s + 5.15·41-s + 1.85·42-s − 0.457·43-s − 0.147·46-s − 0.875·47-s − 2.42·49-s − 1.68·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(1.12634\times 10^{7}\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8886435654\)
\(L(\frac12)\) \(\approx\) \(0.8886435654\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( 1 \)
good2 \( 1 - T + T^{2} - p T^{3} + 3 T^{4} - 7 T^{5} + 11 T^{6} - 7 p T^{7} + 3 p^{2} T^{8} - p^{4} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 3 p T^{2} + 4 p T^{3} + 248 T^{4} + 258 T^{5} + 2000 T^{6} + 258 p T^{7} + 248 p^{2} T^{8} + 4 p^{4} T^{9} + 3 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 37 T^{2} - 8 T^{3} + 723 T^{4} - 168 T^{5} + 9470 T^{6} - 168 p T^{7} + 723 p^{2} T^{8} - 8 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 22 T^{2} + 74 T^{3} + 447 T^{4} + 874 T^{5} + 8929 T^{6} + 874 p T^{7} + 447 p^{2} T^{8} + 74 p^{3} T^{9} + 22 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 2 T + 47 T^{2} - 90 T^{3} + 75 p T^{4} - 2852 T^{5} + 25434 T^{6} - 2852 p T^{7} + 75 p^{3} T^{8} - 90 p^{3} T^{9} + 47 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 40 T^{2} + 70 T^{3} + 1201 T^{4} + 1530 T^{5} + 24751 T^{6} + 1530 p T^{7} + 1201 p^{2} T^{8} + 70 p^{3} T^{9} + 40 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + T + 3 p T^{2} - 29 T^{3} + 2123 T^{4} - 4406 T^{5} + 48270 T^{6} - 4406 p T^{7} + 2123 p^{2} T^{8} - 29 p^{3} T^{9} + 3 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 31 T + 553 T^{2} - 235 p T^{3} + 63999 T^{4} - 474070 T^{5} + 2837054 T^{6} - 474070 p T^{7} + 63999 p^{2} T^{8} - 235 p^{4} T^{9} + 553 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 100 T^{2} + 338 T^{3} + 5477 T^{4} + 17234 T^{5} + 210111 T^{6} + 17234 p T^{7} + 5477 p^{2} T^{8} + 338 p^{3} T^{9} + 100 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 22 T + 281 T^{2} + 2288 T^{3} + 13168 T^{4} + 55738 T^{5} + 260260 T^{6} + 55738 p T^{7} + 13168 p^{2} T^{8} + 2288 p^{3} T^{9} + 281 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 33 T + 645 T^{2} - 8957 T^{3} + 96287 T^{4} - 830106 T^{5} + 5865606 T^{6} - 830106 p T^{7} + 96287 p^{2} T^{8} - 8957 p^{3} T^{9} + 645 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 182 T^{2} + 624 T^{3} + 15607 T^{4} + 52904 T^{5} + 826891 T^{6} + 52904 p T^{7} + 15607 p^{2} T^{8} + 624 p^{3} T^{9} + 182 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + 5007 T^{4} + 26388 T^{5} + 325404 T^{6} + 26388 p T^{7} + 5007 p^{2} T^{8} + 6 p^{4} T^{9} + 98 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 154 T^{2} - 654 T^{3} + 823 T^{4} + 55684 T^{5} - 465780 T^{6} + 55684 p T^{7} + 823 p^{2} T^{8} - 654 p^{3} T^{9} + 154 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 8 T + 285 T^{2} + 1760 T^{3} + 36611 T^{4} + 179960 T^{5} + 2743806 T^{6} + 179960 p T^{7} + 36611 p^{2} T^{8} + 1760 p^{3} T^{9} + 285 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 34 T + 772 T^{2} - 12098 T^{3} + 152713 T^{4} - 1545558 T^{5} + 13267723 T^{6} - 1545558 p T^{7} + 152713 p^{2} T^{8} - 12098 p^{3} T^{9} + 772 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 2 T + 292 T^{2} - 130 T^{3} + 37645 T^{4} + 17358 T^{5} + 3024439 T^{6} + 17358 p T^{7} + 37645 p^{2} T^{8} - 130 p^{3} T^{9} + 292 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 3 T + 201 T^{2} + 1225 T^{3} + 17495 T^{4} + 189438 T^{5} + 1160814 T^{6} + 189438 p T^{7} + 17495 p^{2} T^{8} + 1225 p^{3} T^{9} + 201 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 36 T + 869 T^{2} + 14736 T^{3} + 202708 T^{4} + 2250494 T^{5} + 21131980 T^{6} + 2250494 p T^{7} + 202708 p^{2} T^{8} + 14736 p^{3} T^{9} + 869 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 25 T + 624 T^{2} - 120 p T^{3} + 137585 T^{4} - 1471760 T^{5} + 14938465 T^{6} - 1471760 p T^{7} + 137585 p^{2} T^{8} - 120 p^{4} T^{9} + 624 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T + 369 T^{2} + 3356 T^{3} + 61283 T^{4} + 441504 T^{5} + 6208854 T^{6} + 441504 p T^{7} + 61283 p^{2} T^{8} + 3356 p^{3} T^{9} + 369 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 315 T^{2} - 3670 T^{3} + 38351 T^{4} - 311400 T^{5} + 3203706 T^{6} - 311400 p T^{7} + 38351 p^{2} T^{8} - 3670 p^{3} T^{9} + 315 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 272 T^{2} - 1810 T^{3} + 40125 T^{4} - 194182 T^{5} + 4458809 T^{6} - 194182 p T^{7} + 40125 p^{2} T^{8} - 1810 p^{3} T^{9} + 272 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.90595801696236677297099577332, −4.63578320481153763565583000844, −4.46145667082736714187708050836, −4.32752492128477389280848744703, −4.31093117470162075439466582475, −4.31075056927060029191835508244, −4.28768954882655308870248742185, −3.75352544513448775842440088679, −3.65897765612115493667733859246, −3.51904496805432168623135122944, −3.36759856488971793483560059971, −2.96680357022576370672699099746, −2.86496542753730561422586226867, −2.78871097894235150836168026867, −2.77986636199752314665114051051, −2.22916478683311507921517424111, −2.05660317791085302023197108228, −1.86173746787456459045869034207, −1.83332300066461520333716165140, −1.36321171176972805673525430710, −1.04569297838411953444942667075, −0.898779809436040667508083546091, −0.843430518818154741270944072067, −0.62956313414414826670431414809, −0.17652298395535700095346826699, 0.17652298395535700095346826699, 0.62956313414414826670431414809, 0.843430518818154741270944072067, 0.898779809436040667508083546091, 1.04569297838411953444942667075, 1.36321171176972805673525430710, 1.83332300066461520333716165140, 1.86173746787456459045869034207, 2.05660317791085302023197108228, 2.22916478683311507921517424111, 2.77986636199752314665114051051, 2.78871097894235150836168026867, 2.86496542753730561422586226867, 2.96680357022576370672699099746, 3.36759856488971793483560059971, 3.51904496805432168623135122944, 3.65897765612115493667733859246, 3.75352544513448775842440088679, 4.28768954882655308870248742185, 4.31075056927060029191835508244, 4.31093117470162075439466582475, 4.32752492128477389280848744703, 4.46145667082736714187708050836, 4.63578320481153763565583000844, 4.90595801696236677297099577332

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

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