L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s − 20-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s − 20-s + ⋯ |
Λ(s)=(=(119s/2ΓR(s)L(s)(0.0633+0.997i)Λ(1−s)
Λ(s)=(=(119s/2ΓR(s)L(s)(0.0633+0.997i)Λ(1−s)
Degree: |
1 |
Conductor: |
119
= 7⋅17
|
Sign: |
0.0633+0.997i
|
Analytic conductor: |
0.552633 |
Root analytic conductor: |
0.552633 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ119(67,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 119, (0: ), 0.0633+0.997i)
|
Particular Values
L(21) |
≈ |
0.7395766865+0.6941307269i |
L(21) |
≈ |
0.7395766865+0.6941307269i |
L(1) |
≈ |
0.8602188261+0.5207597101i |
L(1) |
≈ |
0.8602188261+0.5207597101i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 17 | 1 |
good | 2 | 1+(−0.5+0.866i)T |
| 3 | 1+(0.5+0.866i)T |
| 5 | 1+(0.5−0.866i)T |
| 11 | 1+(0.5+0.866i)T |
| 13 | 1+T |
| 19 | 1+(−0.5+0.866i)T |
| 23 | 1+(0.5−0.866i)T |
| 29 | 1−T |
| 31 | 1+(0.5+0.866i)T |
| 37 | 1+(0.5−0.866i)T |
| 41 | 1−T |
| 43 | 1+T |
| 47 | 1+(−0.5+0.866i)T |
| 53 | 1+(−0.5−0.866i)T |
| 59 | 1+(−0.5−0.866i)T |
| 61 | 1+(0.5−0.866i)T |
| 67 | 1+(−0.5−0.866i)T |
| 71 | 1−T |
| 73 | 1+(0.5+0.866i)T |
| 79 | 1+(0.5−0.866i)T |
| 83 | 1+T |
| 89 | 1+(−0.5+0.866i)T |
| 97 | 1−T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−29.21352693684699365352288523441, −28.01037664494439832085969731531, −26.753955652404654590962983548852, −25.92378842199122893546869726636, −25.23156283638037771000134262002, −23.79134498427505655352792782019, −22.58872532040404753402337511559, −21.56434928043545891462760609513, −20.57773908508357004485497349251, −19.34356073672358427422599908252, −18.77270954268158014243800505459, −17.88252760388448585100273782276, −16.92037394095974543375936708780, −15.05949969492908467145742915545, −13.636842044796262635610362793580, −13.31353398626273478158716649955, −11.66858642817689055918792259580, −10.93507412116790114210820391274, −9.45012958284800247737613853050, −8.52714700537861474363421792085, −7.25912092777400212022992478753, −6.07270750520787274395236668747, −3.66589884283648345597505083171, −2.67987721724846746448847029356, −1.34778154864226548141337533994,
1.72343709506298242277298864133, 4.06336415794236928222194805320, 5.088571447260442331761534517238, 6.32515620702562396511474275336, 8.0228119815987259708981482976, 8.92780775820332825082932802594, 9.70984004502317080397096776431, 10.80052495940216961686188340466, 12.7801492578387912917980449755, 14.0004149386233232042917699408, 14.87730353189799850592534791315, 16.007377636804515270349978839491, 16.73164061981212339925836497502, 17.69244611453500009603298970502, 19.059796090544968681791214013928, 20.26242571428869999097653325650, 20.965336878504876807103651482634, 22.38013802270856559447817090866, 23.36924727941291260403273151209, 24.79455045854034238095662090382, 25.33802470099591904149056013170, 26.18788680171413776400032699257, 27.376956126624977926167258108268, 28.055792454792376810687798141652, 28.833312597842647303656157819745