L(s) = 1 | + 2·2-s − 9·3-s + 5·4-s − 18·6-s − 10·7-s − 4·8-s + 54·9-s + 33·11-s − 45·12-s − 114·13-s − 20·14-s − 27·16-s + 104·17-s + 108·18-s − 58·19-s + 90·21-s + 66·22-s − 120·23-s + 36·24-s − 228·26-s − 270·27-s − 50·28-s − 220·29-s + 248·31-s + 6·32-s − 297·33-s + 208·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 5/8·4-s − 1.22·6-s − 0.539·7-s − 0.176·8-s + 2·9-s + 0.904·11-s − 1.08·12-s − 2.43·13-s − 0.381·14-s − 0.421·16-s + 1.48·17-s + 1.41·18-s − 0.700·19-s + 0.935·21-s + 0.639·22-s − 1.08·23-s + 0.306·24-s − 1.71·26-s − 1.92·27-s − 0.337·28-s − 1.40·29-s + 1.43·31-s + 0.0331·32-s − 1.56·33-s + 1.04·34-s + ⋯ |
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s)3L(s)−Λ(4−s)
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s+3/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
33⋅56⋅113
|
Sign: |
−1
|
Analytic conductor: |
115334. |
Root analytic conductor: |
6.97686 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 33⋅56⋅113, ( :3/2,3/2,3/2), −1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | C1 | (1+pT)3 |
| 5 | | 1 |
| 11 | C1 | (1−pT)3 |
good | 2 | S4×C2 | 1−pT−T2+p4T3−p3T4−p7T5+p9T6 |
| 7 | S4×C2 | 1+10T+425T2+3412T3+425p3T4+10p6T5+p9T6 |
| 13 | S4×C2 | 1+114T+10303T2+538132T3+10303p3T4+114p6T5+p9T6 |
| 17 | S4×C2 | 1−104T+17531T2−1030352T3+17531p3T4−104p6T5+p9T6 |
| 19 | S4×C2 | 1+58T+9081T2+861164T3+9081p3T4+58p6T5+p9T6 |
| 23 | S4×C2 | 1+120T+25765T2+2771856T3+25765p3T4+120p6T5+p9T6 |
| 29 | S4×C2 | 1+220T+58859T2+10101400T3+58859p3T4+220p6T5+p9T6 |
| 31 | S4×C2 | 1−8pT+50781T2−5187088T3+50781p3T4−8p7T5+p9T6 |
| 37 | S4×C2 | 1+838T+375507T2+103501764T3+375507p3T4+838p6T5+p9T6 |
| 41 | S4×C2 | 1−156T+100167T2−24516984T3+100167p3T4−156p6T5+p9T6 |
| 43 | S4×C2 | 1+122T+193581T2+17954308T3+193581p3T4+122p6T5+p9T6 |
| 47 | S4×C2 | 1+504T+393661T2+109025808T3+393661p3T4+504p6T5+p9T6 |
| 53 | S4×C2 | 1+282T+224563T2+87620892T3+224563p3T4+282p6T5+p9T6 |
| 59 | S4×C2 | 1−548T+11419pT2−226302104T3+11419p4T4−548p6T5+p9T6 |
| 61 | S4×C2 | 1−414T−389T2+154404524T3−389p3T4−414p6T5+p9T6 |
| 67 | S4×C2 | 1−428T+695537T2−249317576T3+695537p3T4−428p6T5+p9T6 |
| 71 | S4×C2 | 1+912T+1171621T2+649961952T3+1171621p3T4+912p6T5+p9T6 |
| 73 | S4×C2 | 1+618T+1066507T2+454366420T3+1066507p3T4+618p6T5+p9T6 |
| 79 | S4×C2 | 1+542T+1317893T2+445950836T3+1317893p3T4+542p6T5+p9T6 |
| 83 | S4×C2 | 1+624021T2+434328048T3+624021p3T4+p9T6 |
| 89 | S4×C2 | 1−790T−3625T2+827778380T3−3625p3T4−790p6T5+p9T6 |
| 97 | S4×C2 | 1+2074T+3705231T2+3687692268T3+3705231p3T4+2074p6T5+p9T6 |
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L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.455059253445627846431972477339, −8.789070329456716328700463922895, −8.507383788613095679588096596369, −8.400916356948033273541867174937, −7.66971650226535703084286455320, −7.59864198458999920060838161953, −7.22476365326484019209834724176, −7.04021622172011983038074682840, −6.67428184122129793160850072834, −6.42590762522689710887739135138, −6.19632369292971837073748416110, −5.94190570289524022226308542413, −5.48066244628059150307027467943, −5.25841283615790482793916750187, −4.95270927244098311351284433210, −4.86201891524306016678154184888, −4.27893732612711307807493969845, −3.98717575804290234756498577318, −3.65529333290895452112174093378, −3.17526088588212173964864198726, −2.96090904958045158740593924008, −2.24488153647382894389430809733, −2.01897951929802681699068491839, −1.46064607255351147116592512346, −1.09068984393183498145687599081, 0, 0, 0,
1.09068984393183498145687599081, 1.46064607255351147116592512346, 2.01897951929802681699068491839, 2.24488153647382894389430809733, 2.96090904958045158740593924008, 3.17526088588212173964864198726, 3.65529333290895452112174093378, 3.98717575804290234756498577318, 4.27893732612711307807493969845, 4.86201891524306016678154184888, 4.95270927244098311351284433210, 5.25841283615790482793916750187, 5.48066244628059150307027467943, 5.94190570289524022226308542413, 6.19632369292971837073748416110, 6.42590762522689710887739135138, 6.67428184122129793160850072834, 7.04021622172011983038074682840, 7.22476365326484019209834724176, 7.59864198458999920060838161953, 7.66971650226535703084286455320, 8.400916356948033273541867174937, 8.507383788613095679588096596369, 8.789070329456716328700463922895, 9.455059253445627846431972477339