L(s) = 1 | + (−1.62 + 1.93i)2-s + (−1.32 + 1.11i)3-s + (−0.766 − 4.34i)4-s − 4.38i·6-s + (−3.01 − 0.532i)7-s + (5.28 + 3.05i)8-s + (0.520 − 2.95i)9-s + (−5.29 + 1.92i)11-s + (5.85 + 4.91i)12-s + (−2.71 − 3.23i)13-s + (5.94 − 4.98i)14-s + (−6.23 + 2.27i)16-s + (−1.43 + 0.826i)17-s + (4.88 + 5.81i)18-s + (0.120 − 0.208i)19-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.37i)2-s + (−0.766 + 0.642i)3-s + (−0.383 − 2.17i)4-s − 1.79i·6-s + (−1.14 − 0.201i)7-s + (1.86 + 1.07i)8-s + (0.173 − 0.984i)9-s + (−1.59 + 0.581i)11-s + (1.68 + 1.41i)12-s + (−0.753 − 0.898i)13-s + (1.58 − 1.33i)14-s + (−1.55 + 0.567i)16-s + (−0.347 + 0.200i)17-s + (1.15 + 1.37i)18-s + (0.0276 − 0.0479i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(0.117−0.993i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(0.117−0.993i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
0.117−0.993i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(124,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), 0.117−0.993i)
|
Particular Values
L(1) |
≈ |
0.229894+0.204223i |
L(21) |
≈ |
0.229894+0.204223i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.32−1.11i)T |
| 5 | 1 |
good | 2 | 1+(1.62−1.93i)T+(−0.347−1.96i)T2 |
| 7 | 1+(3.01+0.532i)T+(6.57+2.39i)T2 |
| 11 | 1+(5.29−1.92i)T+(8.42−7.07i)T2 |
| 13 | 1+(2.71+3.23i)T+(−2.25+12.8i)T2 |
| 17 | 1+(1.43−0.826i)T+(8.5−14.7i)T2 |
| 19 | 1+(−0.120+0.208i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−7.34+1.29i)T+(21.6−7.86i)T2 |
| 29 | 1+(−5.90−4.95i)T+(5.03+28.5i)T2 |
| 31 | 1+(−0.858−4.86i)T+(−29.1+10.6i)T2 |
| 37 | 1+(2.15−1.24i)T+(18.5−32.0i)T2 |
| 41 | 1+(0.109−0.0918i)T+(7.11−40.3i)T2 |
| 43 | 1+(0.256+0.705i)T+(−32.9+27.6i)T2 |
| 47 | 1+(−4.58−0.807i)T+(44.1+16.0i)T2 |
| 53 | 1+12.1iT−53T2 |
| 59 | 1+(4.45+1.62i)T+(45.1+37.9i)T2 |
| 61 | 1+(−2.41+13.6i)T+(−57.3−20.8i)T2 |
| 67 | 1+(−4.73−5.64i)T+(−11.6+65.9i)T2 |
| 71 | 1+(−2.45−4.24i)T+(−35.5+61.4i)T2 |
| 73 | 1+(0.196+0.113i)T+(36.5+63.2i)T2 |
| 79 | 1+(−7.53−6.32i)T+(13.7+77.7i)T2 |
| 83 | 1+(5.69−6.78i)T+(−14.4−81.7i)T2 |
| 89 | 1+(−3.33+5.76i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−3.26−8.95i)T+(−74.3+62.3i)T2 |
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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
−10.27723032272631887850413285168, −9.902857940611790602294225956561, −8.998094026680213380479084939593, −8.052742123561468298035017568349, −6.98221638394586577157883481173, −6.60233447607412222973853100112, −5.29598526685232637178216815050, −4.99160838798132816628786242078, −3.05328623206781612081605159449, −0.49223401117379135420759553065,
0.61989409660249548682358214669, 2.32169170507917870786312608845, 2.97443540934827751570052927838, 4.62004369653169846911691542393, 5.91622434183180780572628840674, 7.07356510508538193149882016141, 7.80039204682444795317381775109, 8.833972687829516890439232318991, 9.641474425286517455523392659099, 10.42879375843901552245798692224