L(s) = 1 | + (−1.15 + 1.99i)2-s + (1.08 − 1.87i)3-s + (−1.65 − 2.86i)4-s − 2.16·5-s + (2.49 + 4.32i)6-s + 2.99·8-s + (−0.848 − 1.46i)9-s + (2.49 − 4.32i)10-s + (−2.45 + 4.25i)11-s − 7.15·12-s + (1.41 − 3.31i)13-s + (−2.34 + 4.06i)15-s + (−0.151 + 0.262i)16-s + (3.57 + 6.19i)17-s + 3.90·18-s + (1.08 + 1.87i)19-s + ⋯ |
L(s) = 1 | + (−0.814 + 1.41i)2-s + (0.625 − 1.08i)3-s + (−0.825 − 1.43i)4-s − 0.969·5-s + (1.01 + 1.76i)6-s + 1.06·8-s + (−0.282 − 0.489i)9-s + (0.789 − 1.36i)10-s + (−0.739 + 1.28i)11-s − 2.06·12-s + (0.391 − 0.920i)13-s + (−0.606 + 1.05i)15-s + (−0.0378 + 0.0655i)16-s + (0.868 + 1.50i)17-s + 0.921·18-s + (0.248 + 0.430i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.367−0.929i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.367−0.929i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.367−0.929i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(295,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.367−0.929i)
|
Particular Values
L(1) |
≈ |
0.448164+0.659225i |
L(21) |
≈ |
0.448164+0.659225i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(−1.41+3.31i)T |
good | 2 | 1+(1.15−1.99i)T+(−1−1.73i)T2 |
| 3 | 1+(−1.08+1.87i)T+(−1.5−2.59i)T2 |
| 5 | 1+2.16T+5T2 |
| 11 | 1+(2.45−4.25i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−3.57−6.19i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−1.08−1.87i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−0.302+0.524i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−1.15+1.99i)T+(−14.5−25.1i)T2 |
| 31 | 1−7.15T+31T2 |
| 37 | 1+(4.30−7.45i)T+(−18.5−32.0i)T2 |
| 41 | 1+(4.99−8.64i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−6.25−10.8i)T+(−21.5+37.2i)T2 |
| 47 | 1−1.51T+47T2 |
| 53 | 1−2.39T+53T2 |
| 59 | 1+(1.41+2.44i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−2.16−3.75i)T+(−30.5+52.8i)T2 |
| 67 | 1+(0.5−0.866i)T+(−33.5−58.0i)T2 |
| 71 | 1+(2+3.46i)T+(−35.5+61.4i)T2 |
| 73 | 1−4.33T+73T2 |
| 79 | 1+6.60T+79T2 |
| 83 | 1+2.82T+83T2 |
| 89 | 1+(−3.25+5.63i)T+(−44.5−77.0i)T2 |
| 97 | 1+(6.83+11.8i)T+(−48.5+84.0i)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
−10.39469348034162513396499151482, −9.829613788433693205944662800233, −8.328687785244360579516862348051, −8.088945163810389734116506738608, −7.64813368552144983755561754417, −6.76181965864321330459666404045, −5.87342808096711618060379949318, −4.60988042769433069862696946943, −3.03620462983589682119790021823, −1.29433465208163791353622359503,
0.60362787116950152001889747997, 2.63077672105226735147720685904, 3.43497354338087349875385977757, 4.07281619270835394406863968169, 5.35580153517349633868254901946, 7.19596858595444057969939025397, 8.266556239537740206000380291653, 8.855536306809945186006042545372, 9.472085181666232473027127419445, 10.39550117063756666459622350137