L(s) = 1 | + (−2.11 + 2.11i)2-s + (−0.751 − 0.751i)3-s − 4.91i·4-s + (−1.47 + 4.77i)5-s + 3.17·6-s + (−3.30 + 3.30i)7-s + (1.92 + 1.92i)8-s − 7.87i·9-s + (−6.96 − 13.2i)10-s − 2.07·11-s + (−3.69 + 3.69i)12-s + (−9.71 − 9.71i)13-s − 13.9i·14-s + (4.70 − 2.48i)15-s + 11.5·16-s + (−2.26 + 2.26i)17-s + ⋯ |
L(s) = 1 | + (−1.05 + 1.05i)2-s + (−0.250 − 0.250i)3-s − 1.22i·4-s + (−0.295 + 0.955i)5-s + 0.528·6-s + (−0.472 + 0.472i)7-s + (0.240 + 0.240i)8-s − 0.874i·9-s + (−0.696 − 1.32i)10-s − 0.188·11-s + (−0.307 + 0.307i)12-s + (−0.747 − 0.747i)13-s − 0.997i·14-s + (0.313 − 0.165i)15-s + 0.720·16-s + (−0.133 + 0.133i)17-s + ⋯ |
Λ(s)=(=(115s/2ΓC(s)L(s)(−0.0679+0.997i)Λ(3−s)
Λ(s)=(=(115s/2ΓC(s+1)L(s)(−0.0679+0.997i)Λ(1−s)
Degree: |
2 |
Conductor: |
115
= 5⋅23
|
Sign: |
−0.0679+0.997i
|
Analytic conductor: |
3.13352 |
Root analytic conductor: |
1.77017 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ115(93,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 115, ( :1), −0.0679+0.997i)
|
Particular Values
L(23) |
≈ |
0.0453026−0.0484911i |
L(21) |
≈ |
0.0453026−0.0484911i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(1.47−4.77i)T |
| 23 | 1+(−3.39−3.39i)T |
good | 2 | 1+(2.11−2.11i)T−4iT2 |
| 3 | 1+(0.751+0.751i)T+9iT2 |
| 7 | 1+(3.30−3.30i)T−49iT2 |
| 11 | 1+2.07T+121T2 |
| 13 | 1+(9.71+9.71i)T+169iT2 |
| 17 | 1+(2.26−2.26i)T−289iT2 |
| 19 | 1+32.3iT−361T2 |
| 29 | 1−21.3iT−841T2 |
| 31 | 1+47.1T+961T2 |
| 37 | 1+(30.9−30.9i)T−1.36e3iT2 |
| 41 | 1+8.65T+1.68e3T2 |
| 43 | 1+(23.0+23.0i)T+1.84e3iT2 |
| 47 | 1+(−4.91+4.91i)T−2.20e3iT2 |
| 53 | 1+(−16.0−16.0i)T+2.80e3iT2 |
| 59 | 1−15.8iT−3.48e3T2 |
| 61 | 1−92.5T+3.72e3T2 |
| 67 | 1+(−2.96+2.96i)T−4.48e3iT2 |
| 71 | 1+112.T+5.04e3T2 |
| 73 | 1+(−4.53−4.53i)T+5.32e3iT2 |
| 79 | 1+17.0iT−6.24e3T2 |
| 83 | 1+(76.6+76.6i)T+6.88e3iT2 |
| 89 | 1−103.iT−7.92e3T2 |
| 97 | 1+(−71.0+71.0i)T−9.40e3iT2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.02084005861265448120258657526, −11.95113123213863465426401468239, −10.68135255018502043752566413433, −9.606016605947787981348976484163, −8.686000090639929884506072245942, −7.26679601011140317337428741826, −6.79213666930022439111352052226, −5.58719811838846120155166116382, −3.13136850485934591586914615576, −0.06592429037849532802622447156,
1.89194620219527878244810521655, 3.91532842591014316289147973827, 5.41756216534105635030705367609, 7.49537016278646516629779532102, 8.499514791334228870636778297998, 9.617087453940746871898052941015, 10.30588386308642189305102098397, 11.38793066681877479480364704789, 12.24998737600441809343677314895, 13.16466211480802301496731520471