L(s) = 1 | + 12·2-s + 22·3-s + 96·4-s + 264·6-s + 234·7-s + 640·8-s − 45·9-s + 48·11-s + 2.11e3·12-s − 507·13-s + 2.80e3·14-s + 3.84e3·16-s − 1.50e3·17-s − 540·18-s − 360·19-s + 5.14e3·21-s + 576·22-s + 2.37e3·23-s + 1.40e4·24-s − 6.08e3·26-s − 2.93e3·27-s + 2.24e4·28-s − 3.07e3·29-s − 5.38e3·31-s + 2.15e4·32-s + 1.05e3·33-s − 1.80e4·34-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.41·3-s + 3·4-s + 2.99·6-s + 1.80·7-s + 3.53·8-s − 0.185·9-s + 0.119·11-s + 4.23·12-s − 0.832·13-s + 3.82·14-s + 15/4·16-s − 1.26·17-s − 0.392·18-s − 0.228·19-s + 2.54·21-s + 0.253·22-s + 0.934·23-s + 4.98·24-s − 1.76·26-s − 0.774·27-s + 5.41·28-s − 0.679·29-s − 1.00·31-s + 3.71·32-s + 0.168·33-s − 2.68·34-s + ⋯ |
Λ(s)=(=((23⋅56⋅133)s/2ΓC(s)3L(s)Λ(6−s)
Λ(s)=(=((23⋅56⋅133)s/2ΓC(s+5/2)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
23⋅56⋅133
|
Sign: |
1
|
Analytic conductor: |
1.13297×106 |
Root analytic conductor: |
10.2102 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 23⋅56⋅133, ( :5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
55.47093913 |
L(21) |
≈ |
55.47093913 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−p2T)3 |
| 5 | | 1 |
| 13 | C1 | (1+p2T)3 |
good | 3 | S4×C2 | 1−22T+529T2−3232pT3+529p5T4−22p10T5+p15T6 |
| 7 | S4×C2 | 1−234T+48705T2−7824532T3+48705p5T4−234p10T5+p15T6 |
| 11 | S4×C2 | 1−48T+76377T2+58976052T3+76377p5T4−48p10T5+p15T6 |
| 17 | S4×C2 | 1+1506T+3273807T2+2997817500T3+3273807p5T4+1506p10T5+p15T6 |
| 19 | S4×C2 | 1+360T+161955pT2−1434259964T3+161955p6T4+360p10T5+p15T6 |
| 23 | S4×C2 | 1−2370T+19159293T2−30182176920T3+19159293p5T4−2370p10T5+p15T6 |
| 29 | S4×C2 | 1+3078T+41304531T2+146324754036T3+41304531p5T4+3078p10T5+p15T6 |
| 31 | S4×C2 | 1+5388T+26264277T2+161901418108T3+26264277p5T4+5388p10T5+p15T6 |
| 37 | S4×C2 | 1−25362T+395488467T2−3869148349420T3+395488467p5T4−25362p10T5+p15T6 |
| 41 | S4×C2 | 1+15906T−72143817T2−3142322038788T3−72143817p5T4+15906p10T5+p15T6 |
| 43 | S4×C2 | 1−39306T+941655657T2−13605136511296T3+941655657p5T4−39306p10T5+p15T6 |
| 47 | S4×C2 | 1−17778T+430606137T2−3892325349540T3+430606137p5T4−17778p10T5+p15T6 |
| 53 | S4×C2 | 1+9246T−139832133T2+2191133695476T3−139832133p5T4+9246p10T5+p15T6 |
| 59 | S4×C2 | 1+77760T+3729319041T2+118237047820380T3+3729319041p5T4+77760p10T5+p15T6 |
| 61 | S4×C2 | 1−17982T+1546877139T2−16008491590916T3+1546877139p5T4−17982p10T5+p15T6 |
| 67 | S4×C2 | 1−2922T+1163644365T2−57141979467316T3+1163644365p5T4−2922p10T5+p15T6 |
| 71 | S4×C2 | 1+4944T+3669025341T2+114486868308T3+3669025341p5T4+4944p10T5+p15T6 |
| 73 | S4×C2 | 1+43278T+5189491839T2+177703264664708T3+5189491839p5T4+43278p10T5+p15T6 |
| 79 | S4×C2 | 1−42120T+5223395229T2−282172460852048T3+5223395229p5T4−42120p10T5+p15T6 |
| 83 | S4×C2 | 1+58098T+10357081509T2+373431668490228T3+10357081509p5T4+58098p10T5+p15T6 |
| 89 | S4×C2 | 1−19614T+11485034871T2−31522212744708T3+11485034871p5T4−19614p10T5+p15T6 |
| 97 | S4×C2 | 1−87078T+14634643503T2−1303843661262292T3+14634643503p5T4−87078p10T5+p15T6 |
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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
−8.765670026982283328815422666282, −8.091084382420908790824892733140, −7.968868778022150678404246625404, −7.76159839654417269503164052501, −7.33961416325794839205208621935, −7.25715286287775521038886666802, −6.85741662499994260570537028487, −6.32779923722187225633952688426, −6.06721758565053115164935135011, −5.80399389139785005565484171032, −5.47222337540423347793952400417, −5.15976472861249016692667227020, −4.54005019225721683972820529774, −4.52782093436346387701863180333, −4.42700831901146259938861051574, −4.07525763361489831936597206476, −3.24190532139064035248684307828, −3.14965351235742196323327859547, −3.02730064186136903564756910865, −2.29995135908343013614979313403, −2.22063973322207767431042394291, −1.99113021213989685235262906358, −1.52453463465547112685543993256, −0.846448305910342452938626606605, −0.50783733117930365376203155698,
0.50783733117930365376203155698, 0.846448305910342452938626606605, 1.52453463465547112685543993256, 1.99113021213989685235262906358, 2.22063973322207767431042394291, 2.29995135908343013614979313403, 3.02730064186136903564756910865, 3.14965351235742196323327859547, 3.24190532139064035248684307828, 4.07525763361489831936597206476, 4.42700831901146259938861051574, 4.52782093436346387701863180333, 4.54005019225721683972820529774, 5.15976472861249016692667227020, 5.47222337540423347793952400417, 5.80399389139785005565484171032, 6.06721758565053115164935135011, 6.32779923722187225633952688426, 6.85741662499994260570537028487, 7.25715286287775521038886666802, 7.33961416325794839205208621935, 7.76159839654417269503164052501, 7.968868778022150678404246625404, 8.091084382420908790824892733140, 8.765670026982283328815422666282