L(s) = 1 | + 3-s − 2·5-s + 3·7-s + 9-s − 7·11-s − 2·15-s + 17-s + 3·19-s + 3·21-s − 16·23-s − 2·25-s − 5·27-s + 29-s + 5·31-s − 7·33-s − 6·35-s + 6·37-s + 41-s + 4·43-s − 2·45-s + 14·47-s + 6·49-s + 51-s − 31·53-s + 14·55-s + 3·57-s − 18·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s − 2.11·11-s − 0.516·15-s + 0.242·17-s + 0.688·19-s + 0.654·21-s − 3.33·23-s − 2/5·25-s − 0.962·27-s + 0.185·29-s + 0.898·31-s − 1.21·33-s − 1.01·35-s + 0.986·37-s + 0.156·41-s + 0.609·43-s − 0.298·45-s + 2.04·47-s + 6/7·49-s + 0.140·51-s − 4.25·53-s + 1.88·55-s + 0.397·57-s − 2.34·59-s + ⋯ |
Λ(s)=(=((218⋅73⋅193)s/2ΓC(s)3L(s)−Λ(2−s)
Λ(s)=(=((218⋅73⋅193)s/2ΓC(s+1/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
218⋅73⋅193
|
Sign: |
−1
|
Analytic conductor: |
313997. |
Root analytic conductor: |
8.24431 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 218⋅73⋅193, ( :1/2,1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 7 | C1 | (1−T)3 |
| 19 | C1 | (1−T)3 |
good | 3 | S4×C2 | 1−T+2pT3−p2T5+p3T6 |
| 5 | S4×C2 | 1+2T+6T2+14T3+6pT4+2p2T5+p3T6 |
| 11 | S4×C2 | 1+7T+34T2+118T3+34pT4+7p2T5+p3T6 |
| 13 | D6 | 1−54T3+p3T6 |
| 17 | S4×C2 | 1−T+36T2−16T3+36pT4−p2T5+p3T6 |
| 23 | S4×C2 | 1+16T+144T2+832T3+144pT4+16p2T5+p3T6 |
| 29 | S4×C2 | 1−T+72T2−40T3+72pT4−p2T5+p3T6 |
| 31 | S4×C2 | 1−5T+86T2−302T3+86pT4−5p2T5+p3T6 |
| 37 | S4×C2 | 1−6T+30T2−282T3+30pT4−6p2T5+p3T6 |
| 41 | S4×C2 | 1−T+42T2+200T3+42pT4−p2T5+p3T6 |
| 43 | S4×C2 | 1−4T+93T2−296T3+93pT4−4p2T5+p3T6 |
| 47 | S4×C2 | 1−14T+150T2−1100T3+150pT4−14p2T5+p3T6 |
| 53 | S4×C2 | 1+31T+470T2+4284T3+470pT4+31p2T5+p3T6 |
| 59 | S4×C2 | 1+18T+246T2+2160T3+246pT4+18p2T5+p3T6 |
| 61 | S4×C2 | 1+26T+398T2+3734T3+398pT4+26p2T5+p3T6 |
| 67 | S4×C2 | 1−T+94T2−110T3+94pT4−p2T5+p3T6 |
| 71 | S4×C2 | 1+12T+168T2+1380T3+168pT4+12p2T5+p3T6 |
| 73 | S4×C2 | 1−19T+330T2−2960T3+330pT4−19p2T5+p3T6 |
| 79 | S4×C2 | 1−4T+201T2−584T3+201pT4−4p2T5+p3T6 |
| 83 | S4×C2 | 1−13T+4T2+1070T3+4pT4−13p2T5+p3T6 |
| 89 | S4×C2 | 1+12T+147T2+704T3+147pT4+12p2T5+p3T6 |
| 97 | S4×C2 | 1+8T+156T2+1906T3+156pT4+8p2T5+p3T6 |
show more | | |
show less | | |
L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.56166279582442477877951290345, −7.31124340262961857695403791835, −6.69375115936026394808955346389, −6.45960544027304772917685919412, −6.17972968032011986455326086597, −6.09031163492901326334643082658, −5.95168812795559649084778501366, −5.49085967809281739481999438835, −5.39260630696215541654706047723, −5.04974659806428064894944946080, −4.84694033177216992738914313969, −4.62162763805082393144840770843, −4.33255549118268647861889763191, −4.06792939439328067284332581115, −3.89038482177280902377693877958, −3.85494620602601227809270512194, −3.26931899823699679777723016490, −2.87033798465278897083328579224, −2.83374658438719505969266525247, −2.69883093044718487810830946938, −2.18215707412620377421419462156, −1.89549392640071417997303602117, −1.69506803068032357291067361311, −1.33328031139999342752319361856, −0.995622444523982185738811758861, 0, 0, 0,
0.995622444523982185738811758861, 1.33328031139999342752319361856, 1.69506803068032357291067361311, 1.89549392640071417997303602117, 2.18215707412620377421419462156, 2.69883093044718487810830946938, 2.83374658438719505969266525247, 2.87033798465278897083328579224, 3.26931899823699679777723016490, 3.85494620602601227809270512194, 3.89038482177280902377693877958, 4.06792939439328067284332581115, 4.33255549118268647861889763191, 4.62162763805082393144840770843, 4.84694033177216992738914313969, 5.04974659806428064894944946080, 5.39260630696215541654706047723, 5.49085967809281739481999438835, 5.95168812795559649084778501366, 6.09031163492901326334643082658, 6.17972968032011986455326086597, 6.45960544027304772917685919412, 6.69375115936026394808955346389, 7.31124340262961857695403791835, 7.56166279582442477877951290345