L(s) = 1 | − i·2-s − 4-s + (−0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 16-s + 17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 16-s + 17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − i·32-s + ⋯ |
Λ(s)=(=(273s/2ΓR(s)L(s)(0.101−0.994i)Λ(1−s)
Λ(s)=(=(273s/2ΓR(s)L(s)(0.101−0.994i)Λ(1−s)
Degree: |
1 |
Conductor: |
273
= 3⋅7⋅13
|
Sign: |
0.101−0.994i
|
Analytic conductor: |
1.26780 |
Root analytic conductor: |
1.26780 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ273(128,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 273, (0: ), 0.101−0.994i)
|
Particular Values
L(21) |
≈ |
0.7107305230−0.6417039896i |
L(21) |
≈ |
0.7107305230−0.6417039896i |
L(1) |
≈ |
0.7924940068−0.4118404395i |
L(1) |
≈ |
0.7924940068−0.4118404395i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 13 | 1 |
good | 2 | 1−iT |
| 5 | 1+(−0.866+0.5i)T |
| 11 | 1+(0.866−0.5i)T |
| 17 | 1+T |
| 19 | 1+(−0.866−0.5i)T |
| 23 | 1+T |
| 29 | 1+(0.5−0.866i)T |
| 31 | 1+(−0.866−0.5i)T |
| 37 | 1−iT |
| 41 | 1+(0.866+0.5i)T |
| 43 | 1+(0.5+0.866i)T |
| 47 | 1+(0.866−0.5i)T |
| 53 | 1+(0.5−0.866i)T |
| 59 | 1+iT |
| 61 | 1+(−0.5+0.866i)T |
| 67 | 1+(0.866−0.5i)T |
| 71 | 1+(−0.866+0.5i)T |
| 73 | 1+(0.866+0.5i)T |
| 79 | 1+(−0.5−0.866i)T |
| 83 | 1−iT |
| 89 | 1+iT |
| 97 | 1+(0.866−0.5i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.59577539565116444057000174347, −25.1479692286369072531726313748, −24.04740529817811034624703043606, −23.351399960804606399522335644, −22.67537580697975674591480182539, −21.552506248762054110320076267, −20.38559021379346014124469200527, −19.3102206080964389989510234641, −18.617898638228491395148297560039, −17.24484826361898936826297020434, −16.73556886444408047326238865765, −15.75326800458616369506264023204, −14.87299846269819511980366840670, −14.14295352656073491364435014915, −12.70610025774601192270802950713, −12.2042027018615107712225327573, −10.70615375009671644654321176095, −9.36474760957036824736352461827, −8.59030340283182704745471685197, −7.574527816024705334536354920194, −6.72925450611900728165810506662, −5.42603577922107471285497202291, −4.41126911283141243519146895720, −3.49154088050765919636596253836, −1.16071415076833191952673519279,
0.86230501687707515125956139783, 2.53569679149283701876700826107, 3.59871905808262628504924160925, 4.42137364277459741276704354770, 5.90100721424199888150211295822, 7.31617398778363814056318295253, 8.43269028820453350701932126275, 9.37318192822661763347108003293, 10.61300593221580174412903862602, 11.32942982324708886455838524734, 12.13161367205411736168563512186, 13.127839816918932378489099488107, 14.34305058654056967092325189282, 14.961580876166651542646267083826, 16.39616302416573217426092562834, 17.36676441269970833571508554322, 18.52044775653810748097508133108, 19.29208211240067793827418058328, 19.75780163828823876328462437243, 21.01907354814892341962728412087, 21.74448054029597305741291819221, 22.811158170920041495705533994648, 23.286865267601523763557713502231, 24.41672803878134242441388814111, 25.75339648369434676967047460899