L(s) = 1 | + (0.254 + 0.967i)2-s + (−0.466 − 0.884i)3-s + (−0.870 + 0.491i)4-s + (0.564 + 0.825i)5-s + (0.736 − 0.676i)6-s + (0.0855 + 0.996i)7-s + (−0.696 − 0.717i)8-s + (−0.564 + 0.825i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)12-s + (−0.516 − 0.856i)13-s + (−0.941 + 0.336i)14-s + (0.466 − 0.884i)15-s + (0.516 − 0.856i)16-s + (−0.736 + 0.676i)17-s + (−0.941 − 0.336i)18-s + ⋯ |
L(s) = 1 | + (0.254 + 0.967i)2-s + (−0.466 − 0.884i)3-s + (−0.870 + 0.491i)4-s + (0.564 + 0.825i)5-s + (0.736 − 0.676i)6-s + (0.0855 + 0.996i)7-s + (−0.696 − 0.717i)8-s + (−0.564 + 0.825i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)12-s + (−0.516 − 0.856i)13-s + (−0.941 + 0.336i)14-s + (0.466 − 0.884i)15-s + (0.516 − 0.856i)16-s + (−0.736 + 0.676i)17-s + (−0.941 − 0.336i)18-s + ⋯ |
Λ(s)=(=(253s/2ΓR(s)L(s)(−0.895+0.445i)Λ(1−s)
Λ(s)=(=(253s/2ΓR(s)L(s)(−0.895+0.445i)Λ(1−s)
Degree: |
1 |
Conductor: |
253
= 11⋅23
|
Sign: |
−0.895+0.445i
|
Analytic conductor: |
1.17492 |
Root analytic conductor: |
1.17492 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ253(129,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 253, (0: ), −0.895+0.445i)
|
Particular Values
L(21) |
≈ |
0.1920603169+0.8166640569i |
L(21) |
≈ |
0.1920603169+0.8166640569i |
L(1) |
≈ |
0.7025452752+0.5320309538i |
L(1) |
≈ |
0.7025452752+0.5320309538i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 23 | 1 |
good | 2 | 1+(0.254+0.967i)T |
| 3 | 1+(−0.466−0.884i)T |
| 5 | 1+(0.564+0.825i)T |
| 7 | 1+(0.0855+0.996i)T |
| 13 | 1+(−0.516−0.856i)T |
| 17 | 1+(−0.736+0.676i)T |
| 19 | 1+(0.198+0.980i)T |
| 29 | 1+(−0.198+0.980i)T |
| 31 | 1+(−0.985−0.170i)T |
| 37 | 1+(−0.610−0.791i)T |
| 41 | 1+(−0.610+0.791i)T |
| 43 | 1+(−0.142+0.989i)T |
| 47 | 1+(−0.809+0.587i)T |
| 53 | 1+(0.921−0.389i)T |
| 59 | 1+(0.974−0.226i)T |
| 61 | 1+(0.897−0.441i)T |
| 67 | 1+(−0.841+0.540i)T |
| 71 | 1+(0.774+0.633i)T |
| 73 | 1+(0.870−0.491i)T |
| 79 | 1+(0.516+0.856i)T |
| 83 | 1+(−0.0285−0.999i)T |
| 89 | 1+(0.142−0.989i)T |
| 97 | 1+(0.0285−0.999i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.97063073775548171862153799876, −24.24969069671730884920392474663, −23.683384707128178058672417504850, −22.52695545707746130347757115090, −21.84711220259107640997951195463, −20.93797454579476264684819793982, −20.35335362094518611397629121674, −19.5574497202697712373014163221, −18.007985113263964903438810250104, −17.23350302312180516268340230665, −16.502164729829045095286893426305, −15.21569801745667897077333721465, −13.95452863487414067265424534277, −13.3750519981563993822431591174, −12.06473748949579975802292309394, −11.28407697126019565311624497339, −10.263768370823174221939698753570, −9.49031545266437929180681807548, −8.73248079886539066263475703303, −6.778137143393768851352604959789, −5.2882792294863444510695703418, −4.64771705268282336375291012296, −3.74766857752777236052198960735, −2.13952463510092059035605268275, −0.560046738150124289762806973594,
1.97071609772684550816768097465, 3.25533030007023221069652132530, 5.192866341897994841597899427012, 5.85738863505724375566286987207, 6.70167574516068577023962929454, 7.697486355467958621830435924182, 8.675405688634991613549961658682, 10.009554049128458728037467204274, 11.305008455812183424635155665211, 12.55335237556973459614020302038, 13.09462406012643863540463468530, 14.389944224189641204072093768300, 14.90506245859024378854578027418, 16.14504337667033311351353146337, 17.26220768834512486553305414778, 18.04188022297712123260814156524, 18.46513127631100040314758687453, 19.5752141212230515292826527417, 21.34499237525415528820145656597, 22.2677225482876644588091526345, 22.62036260357891458992956984456, 23.779703035195361962512789486597, 24.70302057517837416152760697189, 25.21517679272633884611214041254, 26.01528584827295065711238533680