L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + ⋯ |
Λ(s)=(=(13s/2ΓR(s)L(s)(0.859+0.511i)Λ(1−s)
Λ(s)=(=(13s/2ΓR(s)L(s)(0.859+0.511i)Λ(1−s)
Degree: |
1 |
Conductor: |
13
|
Sign: |
0.859+0.511i
|
Analytic conductor: |
0.0603717 |
Root analytic conductor: |
0.0603717 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ13(4,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 13, (0: ), 0.859+0.511i)
|
Particular Values
L(21) |
≈ |
0.5439933202+0.1495074800i |
L(21) |
≈ |
0.5439933202+0.1495074800i |
L(1) |
≈ |
0.8231273762+0.1859280777i |
L(1) |
≈ |
0.8231273762+0.1859280777i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(0.5+0.866i)T |
| 3 | 1+(−0.5−0.866i)T |
| 5 | 1−T |
| 7 | 1+(0.5−0.866i)T |
| 11 | 1+(0.5+0.866i)T |
| 17 | 1+(−0.5+0.866i)T |
| 19 | 1+(0.5−0.866i)T |
| 23 | 1+(−0.5−0.866i)T |
| 29 | 1+(−0.5−0.866i)T |
| 31 | 1−T |
| 37 | 1+(0.5+0.866i)T |
| 41 | 1+(0.5+0.866i)T |
| 43 | 1+(−0.5+0.866i)T |
| 47 | 1−T |
| 53 | 1+T |
| 59 | 1+(0.5−0.866i)T |
| 61 | 1+(−0.5+0.866i)T |
| 67 | 1+(0.5+0.866i)T |
| 71 | 1+(0.5−0.866i)T |
| 73 | 1−T |
| 79 | 1+T |
| 83 | 1−T |
| 89 | 1+(0.5+0.866i)T |
| 97 | 1+(0.5−0.866i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−44.042538338572824817320349453367, −42.4381505634629781486590400547, −40.54253789223337291520536445179, −39.67695817814093113780009353001, −38.34988310323908670565844503340, −37.54058535341490773551060114884, −35.24321469346156156202293284029, −33.70972897257318246712380632676, −32.04372695737331039705868433686, −31.15408393811924641483439120288, −29.276524014363776500781207107028, −27.784285780986644537584886981993, −27.104781173085398003662802468647, −24.17186681129930729002901535625, −22.68077176274483011651309367952, −21.57530316919110316790947925575, −20.09754801517745622026998311215, −18.382979911139294498264833654699, −15.96780220625727028475050807624, −14.56208942822243278659626643472, −11.99504392637782327338231052927, −11.12452605047154763029618405683, −9.04699334806704048478002090503, −5.42273850072170533908335115787, −3.66097464838443443947257510998,
4.454853911564068083703595038243, 6.80983106916894389432472660976, 7.99512842484801144906191271642, 11.53031305636338257646752187580, 13.03559042617913789432092819792, 14.77690032158991012554008142959, 16.64931851786381053385281175531, 17.93154649566368497638434173782, 19.95742714793315041173639702469, 22.46606086380455732657855395522, 23.56721734639499833998388243743, 24.47347955260311207992017331125, 26.34106658141893115588573031050, 27.94708686509299663405920599574, 30.26856008929587796877185064518, 30.92121120986112335976162807961, 32.88015347184511555416500223370, 34.278212085285161671628017139637, 35.37863743350305940833066011129, 36.39790260310456600503519815589, 39.158077373401764597590580905046, 40.12967928685814949328369040309, 41.47132855075354572950992974996, 42.67809326968208595249856242977, 43.70125476149517110116121945808