L(s) = 1 | + (0.743 + 0.669i)2-s + (−0.891 − 0.453i)3-s + (0.104 + 0.994i)4-s + (0.933 − 0.358i)5-s + (−0.358 − 0.933i)6-s + (−0.838 − 0.544i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.933 + 0.358i)10-s + (−0.707 − 0.707i)11-s + (0.358 − 0.933i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + ⋯ |
L(s) = 1 | + (0.743 + 0.669i)2-s + (−0.891 − 0.453i)3-s + (0.104 + 0.994i)4-s + (0.933 − 0.358i)5-s + (−0.358 − 0.933i)6-s + (−0.838 − 0.544i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.933 + 0.358i)10-s + (−0.707 − 0.707i)11-s + (0.358 − 0.933i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + ⋯ |
Λ(s)=(=(1037s/2ΓR(s)L(s)(0.945+0.324i)Λ(1−s)
Λ(s)=(=(1037s/2ΓR(s)L(s)(0.945+0.324i)Λ(1−s)
Degree: |
1 |
Conductor: |
1037
= 17⋅61
|
Sign: |
0.945+0.324i
|
Analytic conductor: |
4.81580 |
Root analytic conductor: |
4.81580 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1037(117,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1037, (0: ), 0.945+0.324i)
|
Particular Values
L(21) |
≈ |
1.718182927+0.2865632419i |
L(21) |
≈ |
1.718182927+0.2865632419i |
L(1) |
≈ |
1.260351256+0.2395672126i |
L(1) |
≈ |
1.260351256+0.2395672126i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1 |
| 61 | 1 |
good | 2 | 1+(0.743+0.669i)T |
| 3 | 1+(−0.891−0.453i)T |
| 5 | 1+(0.933−0.358i)T |
| 7 | 1+(−0.838−0.544i)T |
| 11 | 1+(−0.707−0.707i)T |
| 13 | 1+(0.5+0.866i)T |
| 19 | 1+(0.207−0.978i)T |
| 23 | 1+(0.987+0.156i)T |
| 29 | 1+(0.258+0.965i)T |
| 31 | 1+(−0.998−0.0523i)T |
| 37 | 1+(0.453+0.891i)T |
| 41 | 1+(0.891−0.453i)T |
| 43 | 1+(0.994+0.104i)T |
| 47 | 1+(0.5−0.866i)T |
| 53 | 1+(0.587−0.809i)T |
| 59 | 1+(−0.743−0.669i)T |
| 67 | 1+(0.913+0.406i)T |
| 71 | 1+(−0.358+0.933i)T |
| 73 | 1+(0.358−0.933i)T |
| 79 | 1+(0.777−0.629i)T |
| 83 | 1+(0.743+0.669i)T |
| 89 | 1+(−0.309−0.951i)T |
| 97 | 1+(0.998+0.0523i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.52412979393543239520814358216, −20.993832632095977639444003081545, −20.33726454939727769501929286285, −19.11854780960888576028281888633, −18.31387234084077748912369087268, −17.90584667447570600013295228795, −16.77188494597814886548047597382, −15.81836234436150195129236992156, −15.25372762879404413803545240503, −14.45526979566606207002243529043, −13.290680044155036553438899883125, −12.73016834711607319130445951598, −12.202986346289819968552336892014, −10.89471684917488756174457869173, −10.58058160044005120954324770637, −9.68557291587705951501213655275, −9.23245364050246765256731226558, −7.39044104873449525148877084303, −6.2743250937998479787743017739, −5.791065357775137546239764382464, −5.2047694647450947065891954108, −4.06634827342548997418721014450, −3.065802947203134225544997903783, −2.26456829379188077881693401646, −0.9514632330455501551793431331,
0.831302108761595192286138145923, 2.24407171337232487728693853482, 3.31413228619406601034136371660, 4.53378391548606608179945313930, 5.31392214083222480099786321873, 6.0123621776402342082939973375, 6.756649474415896293432258157274, 7.32430107233192711651663309124, 8.62112032454476742322155776327, 9.42893409963430890321601964167, 10.692150787768223154537270735931, 11.26428789235968479578463612283, 12.447398920409357642691494991784, 13.0997349307573459082121258786, 13.50172910967886982649316576101, 14.20082164683570372733397552865, 15.58398947672334735599263394673, 16.36362650482564827518680777511, 16.66048301256343390301491583666, 17.501103636642092002036214129023, 18.23437324376620757454616944331, 19.079934354563441647101556974704, 20.25836818344980490245133800016, 21.209525950784442147596727531202, 21.78759686161278996135986830027