L(s) = 1 | + (−0.941 − 0.336i)2-s + (−0.921 + 0.389i)3-s + (0.774 + 0.633i)4-s + (−0.696 − 0.717i)5-s + (0.998 − 0.0570i)6-s + (0.993 + 0.113i)7-s + (−0.516 − 0.856i)8-s + (0.696 − 0.717i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)12-s + (−0.198 + 0.980i)13-s + (−0.897 − 0.441i)14-s + (0.921 + 0.389i)15-s + (0.198 + 0.980i)16-s + (−0.998 + 0.0570i)17-s + (−0.897 + 0.441i)18-s + ⋯ |
L(s) = 1 | + (−0.941 − 0.336i)2-s + (−0.921 + 0.389i)3-s + (0.774 + 0.633i)4-s + (−0.696 − 0.717i)5-s + (0.998 − 0.0570i)6-s + (0.993 + 0.113i)7-s + (−0.516 − 0.856i)8-s + (0.696 − 0.717i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)12-s + (−0.198 + 0.980i)13-s + (−0.897 − 0.441i)14-s + (0.921 + 0.389i)15-s + (0.198 + 0.980i)16-s + (−0.998 + 0.0570i)17-s + (−0.897 + 0.441i)18-s + ⋯ |
Λ(s)=(=(253s/2ΓR(s)L(s)(0.557−0.830i)Λ(1−s)
Λ(s)=(=(253s/2ΓR(s)L(s)(0.557−0.830i)Λ(1−s)
Degree: |
1 |
Conductor: |
253
= 11⋅23
|
Sign: |
0.557−0.830i
|
Analytic conductor: |
1.17492 |
Root analytic conductor: |
1.17492 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ253(112,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 253, (0: ), 0.557−0.830i)
|
Particular Values
L(21) |
≈ |
0.4444939989−0.2368931183i |
L(21) |
≈ |
0.4444939989−0.2368931183i |
L(1) |
≈ |
0.5182272202−0.1006617617i |
L(1) |
≈ |
0.5182272202−0.1006617617i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 23 | 1 |
good | 2 | 1+(−0.941−0.336i)T |
| 3 | 1+(−0.921+0.389i)T |
| 5 | 1+(−0.696−0.717i)T |
| 7 | 1+(0.993+0.113i)T |
| 13 | 1+(−0.198+0.980i)T |
| 17 | 1+(−0.998+0.0570i)T |
| 19 | 1+(−0.254−0.967i)T |
| 29 | 1+(0.254−0.967i)T |
| 31 | 1+(0.974−0.226i)T |
| 37 | 1+(0.985−0.170i)T |
| 41 | 1+(0.985+0.170i)T |
| 43 | 1+(−0.654−0.755i)T |
| 47 | 1+(0.309−0.951i)T |
| 53 | 1+(0.870−0.491i)T |
| 59 | 1+(−0.736+0.676i)T |
| 61 | 1+(0.0855−0.996i)T |
| 67 | 1+(0.959−0.281i)T |
| 71 | 1+(0.610−0.791i)T |
| 73 | 1+(−0.774−0.633i)T |
| 79 | 1+(0.198−0.980i)T |
| 83 | 1+(−0.466−0.884i)T |
| 89 | 1+(0.654+0.755i)T |
| 97 | 1+(0.466−0.884i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−26.399333510546265958101910814634, −25.05533133759419883537112214956, −24.39592423200280171654584494406, −23.476452805057712438250868548948, −22.80696920083954302691516018594, −21.61099737995682611687253060845, −20.30817276059292898452938778340, −19.42772203043109778825623345559, −18.36688840584728303868367206650, −17.916903139696807207537642472058, −17.06641794811139975894500854194, −15.971144575343834300097863875327, −15.15546959297203412086709994958, −14.19348916919861240161809621109, −12.5325459758723422168515084638, −11.4817255829233711368393205065, −10.88313308705619770309507791640, −10.116510351996162831639829959484, −8.35119535598958533718879098240, −7.681715403714577724425305116417, −6.77643834786717579569475963274, −5.74150862754467520106058469549, −4.49775980105280364683054743291, −2.52910562671814518793344617288, −1.088817445725113206897980941313,
0.67326423472921589416673848027, 2.098591271676313581345153172922, 4.08891587291468085627514561472, 4.80049815357918061374477243463, 6.38967619538620453052273165101, 7.48728331646452397612122788776, 8.619904792077548986446316914532, 9.4048959570967624686268016556, 10.72779454380605184872874299812, 11.56125530380830791674368776318, 11.93501801997421306601394229005, 13.22473056881299988249153593499, 15.12151728979594848297226728272, 15.768094340784601104358485580415, 16.806833503395067551625068004415, 17.375297458218031068717169075000, 18.28278653884481773664015350451, 19.35904937729505657592142936505, 20.30789853052355853904269663266, 21.24089403997672871287371564020, 21.807639116285941857628085587536, 23.25341354593992338463345559032, 24.20314545604607287319258792680, 24.673171474736250712590464484596, 26.4077622229093496875205575983