L(s) = 1 | + (0.998 + 0.0448i)2-s + (0.0896 + 0.995i)3-s + (0.995 + 0.0896i)4-s + (0.963 − 0.266i)5-s + (0.0448 + 0.998i)6-s + (0.858 + 0.512i)7-s + (0.990 + 0.134i)8-s + (−0.983 + 0.178i)9-s + (0.974 − 0.222i)10-s + i·12-s + (−0.473 + 0.880i)13-s + (0.834 + 0.550i)14-s + (0.351 + 0.936i)15-s + (0.983 + 0.178i)16-s + (−0.587 + 0.809i)17-s + (−0.990 + 0.134i)18-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0448i)2-s + (0.0896 + 0.995i)3-s + (0.995 + 0.0896i)4-s + (0.963 − 0.266i)5-s + (0.0448 + 0.998i)6-s + (0.858 + 0.512i)7-s + (0.990 + 0.134i)8-s + (−0.983 + 0.178i)9-s + (0.974 − 0.222i)10-s + i·12-s + (−0.473 + 0.880i)13-s + (0.834 + 0.550i)14-s + (0.351 + 0.936i)15-s + (0.983 + 0.178i)16-s + (−0.587 + 0.809i)17-s + (−0.990 + 0.134i)18-s + ⋯ |
Λ(s)=(=(319s/2ΓR(s+1)L(s)(0.0834+0.996i)Λ(1−s)
Λ(s)=(=(319s/2ΓR(s+1)L(s)(0.0834+0.996i)Λ(1−s)
Degree: |
1 |
Conductor: |
319
= 11⋅29
|
Sign: |
0.0834+0.996i
|
Analytic conductor: |
34.2813 |
Root analytic conductor: |
34.2813 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ319(137,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 319, (1: ), 0.0834+0.996i)
|
Particular Values
L(21) |
≈ |
3.731343691+3.431831482i |
L(21) |
≈ |
3.731343691+3.431831482i |
L(1) |
≈ |
2.304852176+1.080338294i |
L(1) |
≈ |
2.304852176+1.080338294i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 29 | 1 |
good | 2 | 1+(0.998+0.0448i)T |
| 3 | 1+(0.0896+0.995i)T |
| 5 | 1+(0.963−0.266i)T |
| 7 | 1+(0.858+0.512i)T |
| 13 | 1+(−0.473+0.880i)T |
| 17 | 1+(−0.587+0.809i)T |
| 19 | 1+(−0.512−0.858i)T |
| 23 | 1+(0.623+0.781i)T |
| 31 | 1+(0.998+0.0448i)T |
| 37 | 1+(−0.722−0.691i)T |
| 41 | 1+(−0.951+0.309i)T |
| 43 | 1+(0.781−0.623i)T |
| 47 | 1+(0.722−0.691i)T |
| 53 | 1+(−0.0448+0.998i)T |
| 59 | 1+(0.309−0.951i)T |
| 61 | 1+(−0.657−0.753i)T |
| 67 | 1+(0.900+0.433i)T |
| 71 | 1+(−0.983−0.178i)T |
| 73 | 1+(0.351+0.936i)T |
| 79 | 1+(−0.178−0.983i)T |
| 83 | 1+(−0.393+0.919i)T |
| 89 | 1+(−0.781−0.623i)T |
| 97 | 1+(−0.657+0.753i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.69437491134433098930045532461, −23.93840976047666792092873858304, −22.816475523516875604910079444748, −22.38089888261628722814306811410, −20.92883620008229345302533033881, −20.60985359307344006068387780143, −19.48035927140154984825765333559, −18.383028262109391317206892154347, −17.45723592222248725778462512870, −16.80259041112760963534846062078, −15.17808080306657754195316923507, −14.364146403711484442459516753949, −13.74785029428701902507799577015, −12.96693423292272161056623907896, −12.05974437976065548768939482486, −11.000549977561499941936677203832, −10.17026536929410159738235841896, −8.47131247140990767074674409254, −7.40012649562419812690444908148, −6.598519464871838589201793657615, −5.607304170005956006915606090808, −4.66630437112825171693852705600, −2.995891083833597268500557044984, −2.15051589397466186099724055696, −1.06064486099966101268531073219,
1.80510377590950276848701994573, 2.664097758457756813102041443, 4.17241373243504755897730867529, 4.937865730811044947440818585507, 5.69266531423891901692609571257, 6.81802423045729301189474413790, 8.42707252226436734240882581746, 9.29254508067150873133950904716, 10.51033602181433630440783739165, 11.276630648967952810120017589028, 12.27851369996036352797881724369, 13.511021778362858507658947154344, 14.220886953958403164901787982410, 15.07707832467931632425340381167, 15.745710341447315148233273205830, 17.11509298531525936625794257144, 17.34383672486142068359779109172, 19.139717654429806242287837639665, 20.21789389538813703166181381714, 21.0768785761789182613463977157, 21.69964927517475484841545145370, 21.96343899957155161861482473822, 23.3410195814540744817058246267, 24.251473152386206562158401521719, 25.026721952895272844369239004833