↑

Properties

Label 12-4014e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.183\times 10^{21}$
Sign $1$
Analytic cond. $1.08424\times 10^{9}$
Root an. cond. $5.66144$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 21·4-s − 6·5-s + 5·7-s − 56·8-s + 36·10-s − 11-s + 6·13-s − 30·14-s + 126·16-s − 10·17-s − 4·19-s − 126·20-s + 6·22-s − 2·23-s + 3·25-s − 36·26-s + 105·28-s − 6·29-s − 5·31-s − 252·32-s + 60·34-s − 30·35-s + 37-s + 24·38-s + 336·40-s − 12·41-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s − 2.68·5-s + 1.88·7-s − 19.7·8-s + 11.3·10-s − 0.301·11-s + 1.66·13-s − 8.01·14-s + 63/2·16-s − 2.42·17-s − 0.917·19-s − 28.1·20-s + 1.27·22-s − 0.417·23-s + 3/5·25-s − 7.06·26-s + 19.8·28-s − 1.11·29-s − 0.898·31-s − 44.5·32-s + 10.2·34-s − 5.07·35-s + 0.164·37-s + 3.89·38-s + 53.1·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 223^{6}\)
Sign: $1$
Analytic conductor: \(1.08424\times 10^{9}\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 223^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
3 \( 1 \)
223 \( ( 1 - T )^{6} \)
good5 \( 1 + 6 T + 33 T^{2} + 121 T^{3} + 408 T^{4} + 1091 T^{5} + 2684 T^{6} + 1091 p T^{7} + 408 p^{2} T^{8} + 121 p^{3} T^{9} + 33 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 5 T + 34 T^{2} - 114 T^{3} + 499 T^{4} - 1345 T^{5} + 4468 T^{6} - 1345 p T^{7} + 499 p^{2} T^{8} - 114 p^{3} T^{9} + 34 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T + 40 T^{2} - 25 T^{3} + 615 T^{4} - 1417 T^{5} + 6516 T^{6} - 1417 p T^{7} + 615 p^{2} T^{8} - 25 p^{3} T^{9} + 40 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 68 T^{2} - 322 T^{3} + 2040 T^{4} - 589 p T^{5} + 34410 T^{6} - 589 p^{2} T^{7} + 2040 p^{2} T^{8} - 322 p^{3} T^{9} + 68 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 10 T + 97 T^{2} + 649 T^{3} + 229 p T^{4} + 18857 T^{5} + 86090 T^{6} + 18857 p T^{7} + 229 p^{3} T^{8} + 649 p^{3} T^{9} + 97 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T + 70 T^{2} + 306 T^{3} + 2570 T^{4} + 10439 T^{5} + 59754 T^{6} + 10439 p T^{7} + 2570 p^{2} T^{8} + 306 p^{3} T^{9} + 70 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T + 65 T^{2} + 235 T^{3} + 2357 T^{4} + 9465 T^{5} + 62842 T^{6} + 9465 p T^{7} + 2357 p^{2} T^{8} + 235 p^{3} T^{9} + 65 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 142 T^{2} + 710 T^{3} + 9120 T^{4} + 36961 T^{5} + 338694 T^{6} + 36961 p T^{7} + 9120 p^{2} T^{8} + 710 p^{3} T^{9} + 142 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 5 T + 126 T^{2} + 530 T^{3} + 7933 T^{4} + 27909 T^{5} + 306992 T^{6} + 27909 p T^{7} + 7933 p^{2} T^{8} + 530 p^{3} T^{9} + 126 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - T + 127 T^{2} + 61 T^{3} + 8406 T^{4} + 5520 T^{5} + 382600 T^{6} + 5520 p T^{7} + 8406 p^{2} T^{8} + 61 p^{3} T^{9} + 127 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 251 T^{2} + 1995 T^{3} + 23819 T^{4} + 141129 T^{5} + 1246146 T^{6} + 141129 p T^{7} + 23819 p^{2} T^{8} + 1995 p^{3} T^{9} + 251 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 11 T + 246 T^{2} + 1877 T^{3} + 24479 T^{4} + 143495 T^{5} + 1364840 T^{6} + 143495 p T^{7} + 24479 p^{2} T^{8} + 1877 p^{3} T^{9} + 246 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 5 T + 204 T^{2} + 1111 T^{3} + 19280 T^{4} + 101170 T^{5} + 1117673 T^{6} + 101170 p T^{7} + 19280 p^{2} T^{8} + 1111 p^{3} T^{9} + 204 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 232 T^{2} + 684 T^{3} + 24888 T^{4} + 57715 T^{5} + 1635504 T^{6} + 57715 p T^{7} + 24888 p^{2} T^{8} + 684 p^{3} T^{9} + 232 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + T + 198 T^{2} + 9 p T^{3} + 16363 T^{4} + 74891 T^{5} + 963204 T^{6} + 74891 p T^{7} + 16363 p^{2} T^{8} + 9 p^{4} T^{9} + 198 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T + 96 T^{2} + 868 T^{3} + 12058 T^{4} + 71629 T^{5} + 630206 T^{6} + 71629 p T^{7} + 12058 p^{2} T^{8} + 868 p^{3} T^{9} + 96 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T + 341 T^{2} + 1697 T^{3} + 51708 T^{4} + 207911 T^{5} + 4466256 T^{6} + 207911 p T^{7} + 51708 p^{2} T^{8} + 1697 p^{3} T^{9} + 341 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + T + 157 T^{2} - 254 T^{3} + 17047 T^{4} - 15675 T^{5} + 1462102 T^{6} - 15675 p T^{7} + 17047 p^{2} T^{8} - 254 p^{3} T^{9} + 157 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 10 T + 249 T^{2} + 2366 T^{3} + 32637 T^{4} + 293431 T^{5} + 2852527 T^{6} + 293431 p T^{7} + 32637 p^{2} T^{8} + 2366 p^{3} T^{9} + 249 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 10 T + 165 T^{2} + 1548 T^{3} + 19319 T^{4} + 124055 T^{5} + 1493471 T^{6} + 124055 p T^{7} + 19319 p^{2} T^{8} + 1548 p^{3} T^{9} + 165 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 6 T + 243 T^{2} - 2429 T^{3} + 38130 T^{4} - 303279 T^{5} + 4244252 T^{6} - 303279 p T^{7} + 38130 p^{2} T^{8} - 2429 p^{3} T^{9} + 243 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T + 212 T^{2} + 432 T^{3} + 27629 T^{4} + 18147 T^{5} + 3437316 T^{6} + 18147 p T^{7} + 27629 p^{2} T^{8} + 432 p^{3} T^{9} + 212 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T + 506 T^{2} - 1466 T^{3} + 113327 T^{4} - 280531 T^{5} + 14292972 T^{6} - 280531 p T^{7} + 113327 p^{2} T^{8} - 1466 p^{3} T^{9} + 506 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.88787312418144223738390710310, −4.64259762020191948879082076615, −4.27842377468447935311547269113, −4.27182428492876514093573159049, −4.07392498316526917684172204602, −4.00188242574905114509137251438, −3.93687861940554450058299876326, −3.69484730185692331005947417030, −3.45546878406090398642879135078, −3.45367358356901074488987639488, −3.31375444670676551741619182032, −3.22284208792312850476082455384, −3.05372633480242391456325037172, −2.59130729687721254575211245552, −2.50690456049711810503374261123, −2.28098069686348481269223581662, −2.22277505025087907388183217747, −2.15910794996260776734890872406, −2.04514349544507725196407362039, −1.55278509342538588158132217028, −1.47888940640797339773904973538, −1.45462496865418571053406543906, −1.36472269448779380220543715044, −1.16990375706232278131204955825, −0.999145651603834286080813506855, 0, 0, 0, 0, 0, 0, 0.999145651603834286080813506855, 1.16990375706232278131204955825, 1.36472269448779380220543715044, 1.45462496865418571053406543906, 1.47888940640797339773904973538, 1.55278509342538588158132217028, 2.04514349544507725196407362039, 2.15910794996260776734890872406, 2.22277505025087907388183217747, 2.28098069686348481269223581662, 2.50690456049711810503374261123, 2.59130729687721254575211245552, 3.05372633480242391456325037172, 3.22284208792312850476082455384, 3.31375444670676551741619182032, 3.45367358356901074488987639488, 3.45546878406090398642879135078, 3.69484730185692331005947417030, 3.93687861940554450058299876326, 4.00188242574905114509137251438, 4.07392498316526917684172204602, 4.27182428492876514093573159049, 4.27842377468447935311547269113, 4.64259762020191948879082076615, 4.88787312418144223738390710310

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

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