L(s) = 1 | + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (0.321 − 0.946i)5-s + (−0.866 + 0.5i)6-s + (0.659 − 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (0.896 + 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (0.946 + 0.321i)13-s + (0.659 + 0.751i)14-s + (0.946 − 0.321i)15-s + (0.866 + 0.5i)16-s + (−0.751 + 0.659i)17-s + ⋯ |
L(s) = 1 | + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (0.321 − 0.946i)5-s + (−0.866 + 0.5i)6-s + (0.659 − 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (0.896 + 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (0.946 + 0.321i)13-s + (0.659 + 0.751i)14-s + (0.946 − 0.321i)15-s + (0.866 + 0.5i)16-s + (−0.751 + 0.659i)17-s + ⋯ |
Λ(s)=(=(97s/2ΓR(s+1)L(s)(0.404+0.914i)Λ(1−s)
Λ(s)=(=(97s/2ΓR(s+1)L(s)(0.404+0.914i)Λ(1−s)
Degree: |
1 |
Conductor: |
97
|
Sign: |
0.404+0.914i
|
Analytic conductor: |
10.4240 |
Root analytic conductor: |
10.4240 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ97(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 97, (1: ), 0.404+0.914i)
|
Particular Values
L(21) |
≈ |
1.782851231+1.160885470i |
L(21) |
≈ |
1.782851231+1.160885470i |
L(1) |
≈ |
1.222134935+0.6317093364i |
L(1) |
≈ |
1.222134935+0.6317093364i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 97 | 1 |
good | 2 | 1+(−0.130+0.991i)T |
| 3 | 1+(0.608+0.793i)T |
| 5 | 1+(0.321−0.946i)T |
| 7 | 1+(0.659−0.751i)T |
| 11 | 1+(0.991−0.130i)T |
| 13 | 1+(0.946+0.321i)T |
| 17 | 1+(−0.751+0.659i)T |
| 19 | 1+(0.980−0.195i)T |
| 23 | 1+(0.0654−0.997i)T |
| 29 | 1+(−0.896+0.442i)T |
| 31 | 1+(0.793−0.608i)T |
| 37 | 1+(0.997−0.0654i)T |
| 41 | 1+(0.442+0.896i)T |
| 43 | 1+(0.258+0.965i)T |
| 47 | 1+(−0.707+0.707i)T |
| 53 | 1+(0.991+0.130i)T |
| 59 | 1+(−0.0654−0.997i)T |
| 61 | 1+(−0.5+0.866i)T |
| 67 | 1+(−0.195−0.980i)T |
| 71 | 1+(0.442−0.896i)T |
| 73 | 1+(−0.965+0.258i)T |
| 79 | 1+(0.923+0.382i)T |
| 83 | 1+(−0.659−0.751i)T |
| 89 | 1+(−0.382+0.923i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−29.82780341441154197011757071779, −28.88068978889254995179815804926, −27.63295227617248852432607990079, −26.63002171434581420841311658074, −25.542675822402090242780303659775, −24.616782758268943259872267433142, −23.0952443144256339725009380635, −22.188775263622861629341434652852, −21.06494591914134672015015486364, −20.05636775078728681236240486656, −18.9728954664883110100722828788, −18.15790092578951937411523327047, −17.588081850142862078370636501631, −15.22351458814624320264276108997, −14.12508590967449341985672318039, −13.42383894461116570170210720349, −11.90313187158717171308485277773, −11.26066989726686212345910025627, −9.59692414659246097104471405464, −8.6641695568007038536417958263, −7.33499068001009755258254054491, −5.76899604241724416001299514184, −3.68824275797015580316197097045, −2.509486383843658485024978740574, −1.374607153179556242951776901012,
1.2520159374244747970118717633, 3.996721200494449740234074645922, 4.70410640799947593036216048047, 6.19695158594891376923627638332, 7.90116419139265405703440704511, 8.80151385608624812322090756421, 9.64700227238350419183450529175, 11.10875251516698363209058682876, 13.19054691089393943290618585692, 13.992883378983600183757232131784, 14.947771951353126669893491854, 16.282242538349268358106627937925, 16.808057238007911709518257718129, 17.9642518880162638751807381940, 19.61491024010848449934431816857, 20.53119932399725101122404504302, 21.62196415727980480832185518358, 22.77452426231541616784681157838, 24.206451391856323064320532639117, 24.72945415428123514630758784831, 25.97366439090387325752693361886, 26.73269629624377589808506790508, 27.75014509522513133588106335392, 28.45890924214028536735448167022, 30.420193982202584269438164053254