L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (−0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (−0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
Λ(s)=(=(19s/2ΓR(s+1)L(s)(0.980+0.196i)Λ(1−s)
Λ(s)=(=(19s/2ΓR(s+1)L(s)(0.980+0.196i)Λ(1−s)
Degree: |
1 |
Conductor: |
19
|
Sign: |
0.980+0.196i
|
Analytic conductor: |
2.04183 |
Root analytic conductor: |
2.04183 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ19(10,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 19, (1: ), 0.980+0.196i)
|
Particular Values
L(21) |
≈ |
1.973817970+0.1957502106i |
L(21) |
≈ |
1.973817970+0.1957502106i |
L(1) |
≈ |
1.660301634+0.1000769021i |
L(1) |
≈ |
1.660301634+0.1000769021i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 19 | 1 |
good | 2 | 1+(0.939−0.342i)T |
| 3 | 1+(−0.173+0.984i)T |
| 5 | 1+(0.766+0.642i)T |
| 7 | 1+(−0.5−0.866i)T |
| 11 | 1+(−0.5+0.866i)T |
| 13 | 1+(−0.173−0.984i)T |
| 17 | 1+(−0.939+0.342i)T |
| 23 | 1+(0.766−0.642i)T |
| 29 | 1+(0.939+0.342i)T |
| 31 | 1+(0.5+0.866i)T |
| 37 | 1−T |
| 41 | 1+(−0.173+0.984i)T |
| 43 | 1+(0.766+0.642i)T |
| 47 | 1+(−0.939−0.342i)T |
| 53 | 1+(−0.766+0.642i)T |
| 59 | 1+(0.939−0.342i)T |
| 61 | 1+(0.766−0.642i)T |
| 67 | 1+(0.939+0.342i)T |
| 71 | 1+(−0.766−0.642i)T |
| 73 | 1+(0.173−0.984i)T |
| 79 | 1+(−0.173+0.984i)T |
| 83 | 1+(−0.5−0.866i)T |
| 89 | 1+(−0.173−0.984i)T |
| 97 | 1+(0.939−0.342i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−40.59294116672808166460496738484, −39.43514244239238140166406361752, −37.59559707179108460393255301060, −35.973527485234823231916095648758, −34.803493837508893892373719786704, −33.56735239520936400902089836715, −31.971763523267286440591611540525, −31.08745948920282231489892560645, −29.30676061222835925079542104486, −28.82642615782907725515631340688, −25.91580962283144924963551746847, −24.763826382247811897420968436766, −23.9715252118784483286726638424, −22.31425436691089341533025009498, −21.071723502346221857968274427526, −19.18057587678369335750470180227, −17.40193068958950159836810892492, −15.98467714326647205240586893365, −13.88679509946100288128298854289, −12.95799316677742842970647875788, −11.635351450120620218563245761694, −8.67354534726236485864325823859, −6.59142772587323881424794900382, −5.37011900372183134500816131601, −2.39276444415460360932038984285,
3.007290648223576261709094943072, 4.844020882450831383706659292615, 6.602885669054097391483674015466, 10.02191643776843284865225332735, 10.74952018439617717345715267411, 12.96375432162960380202140802186, 14.46497014729291701213093981024, 15.707003782804773143765595485240, 17.4947600170617978376102149037, 19.887679113267158044339210294658, 21.03384680967809964075526831563, 22.36670876315206745340868276411, 23.121524521330731448722189255957, 25.30351526184014801392532936517, 26.60139665133819495386195027845, 28.47610320248521675237576170891, 29.54255930682118577613500707697, 30.95069409726045263827618158922, 32.665089019440368273397279488361, 33.15017263515050345128282507175, 34.39380163641819383680604948446, 36.87537487458207336883900507342, 38.03673137821961937974907766312, 39.0901803465859986416684212495, 40.0739957719464559867467688341