L(s) = 1 | + (−0.591 + 0.806i)2-s + (0.116 + 0.993i)3-s + (−0.300 − 0.953i)4-s + (0.960 − 0.277i)5-s + (−0.869 − 0.493i)6-s + (−0.513 − 0.858i)7-s + (0.946 + 0.322i)8-s + (−0.972 + 0.232i)9-s + (−0.344 + 0.938i)10-s + (−0.472 − 0.881i)11-s + (0.912 − 0.409i)12-s + (0.344 − 0.938i)13-s + (0.995 + 0.0936i)14-s + (0.388 + 0.921i)15-s + (−0.819 + 0.572i)16-s + (0.472 − 0.881i)17-s + ⋯ |
L(s) = 1 | + (−0.591 + 0.806i)2-s + (0.116 + 0.993i)3-s + (−0.300 − 0.953i)4-s + (0.960 − 0.277i)5-s + (−0.869 − 0.493i)6-s + (−0.513 − 0.858i)7-s + (0.946 + 0.322i)8-s + (−0.972 + 0.232i)9-s + (−0.344 + 0.938i)10-s + (−0.472 − 0.881i)11-s + (0.912 − 0.409i)12-s + (0.344 − 0.938i)13-s + (0.995 + 0.0936i)14-s + (0.388 + 0.921i)15-s + (−0.819 + 0.572i)16-s + (0.472 − 0.881i)17-s + ⋯ |
Λ(s)=(=(269s/2ΓR(s)L(s)(0.986+0.165i)Λ(1−s)
Λ(s)=(=(269s/2ΓR(s)L(s)(0.986+0.165i)Λ(1−s)
Degree: |
1 |
Conductor: |
269
|
Sign: |
0.986+0.165i
|
Analytic conductor: |
1.24923 |
Root analytic conductor: |
1.24923 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ269(199,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 269, (0: ), 0.986+0.165i)
|
Particular Values
L(21) |
≈ |
0.9071759229+0.07544535978i |
L(21) |
≈ |
0.9071759229+0.07544535978i |
L(1) |
≈ |
0.8197639150+0.2342779810i |
L(1) |
≈ |
0.8197639150+0.2342779810i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 269 | 1 |
good | 2 | 1+(−0.591+0.806i)T |
| 3 | 1+(0.116+0.993i)T |
| 5 | 1+(0.960−0.277i)T |
| 7 | 1+(−0.513−0.858i)T |
| 11 | 1+(−0.472−0.881i)T |
| 13 | 1+(0.344−0.938i)T |
| 17 | 1+(0.472−0.881i)T |
| 19 | 1+(0.209+0.977i)T |
| 23 | 1+(−0.116−0.993i)T |
| 29 | 1+(−0.845−0.533i)T |
| 31 | 1+(0.553−0.833i)T |
| 37 | 1+(−0.628−0.777i)T |
| 41 | 1+(0.591+0.806i)T |
| 43 | 1+(0.845+0.533i)T |
| 47 | 1+(0.163+0.986i)T |
| 53 | 1+(−0.0234−0.999i)T |
| 59 | 1+(0.300+0.953i)T |
| 61 | 1+(0.255+0.966i)T |
| 67 | 1+(−0.300+0.953i)T |
| 71 | 1+(−0.995+0.0936i)T |
| 73 | 1+(0.982+0.186i)T |
| 79 | 1+(−0.990−0.140i)T |
| 83 | 1+(−0.892+0.451i)T |
| 89 | 1+(0.731+0.681i)T |
| 97 | 1+(0.255−0.966i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.8428603133068236739584938011, −25.23818346005540214576767220539, −23.9963940976442645272428382327, −22.84962658414612584711582765432, −21.88868587483187372089227197076, −21.17296182398973999232608344411, −20.08760733394982022705243357229, −19.07477565475323127085333269201, −18.555737009524696769230403081500, −17.68265325396364621803859962713, −17.06122606160546494850878499906, −15.589125123202081697714390618248, −14.17595597075655754486768867821, −13.26352744407788203684185380994, −12.567268654535154555008464231264, −11.70201934429580379543229569183, −10.520548520959553131120583111589, −9.3880614061829454120848827075, −8.78878074293241650365412619283, −7.42074213933108623146870437632, −6.52294599097868048026106297810, −5.27764004435506030866746535097, −3.3451670991737547721616148291, −2.26977498801456817474645536266, −1.59607595796998008874389217380,
0.774994579223579204389035367273, 2.80767089442321253034317781682, 4.25665098222938459919068203702, 5.55821878314840511944663027859, 6.03752697588421019816937144873, 7.640199968641919378527473714855, 8.600184512024585288794767936592, 9.691176913157845471709673065249, 10.19816450912317370023330107763, 11.04015640238328174537844834825, 13.077526115933264004512106157729, 13.919398019479437588841818699360, 14.63879573540897738763112486746, 16.043455512164590865475039406727, 16.37183024685441877176486974742, 17.2280753164401935871648453496, 18.2037309265672419913230120088, 19.297620128261048722302917929437, 20.5547495302188207132819645102, 20.900746919974623742593504408251, 22.55493236613027691022397670261, 22.803221477418266503526398369367, 24.25610976107433455212833955813, 25.09792958175025062122822996739, 25.9425522059111824907689252196