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) |
≈ |
0.269291−0.183073i |
L(21) |
≈ |
0.269291−0.183073i |
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
−10.83614840734325901527247090856, −9.978520127920313271985474553002, −9.487778786814506818322621036381, −8.994608713065853507283245242052, −7.53544377308828971296252325705, −6.92498480371723243229774325658, −5.94202421540598208808864746187, −5.02103384332134758367868695578, −4.56728148578927855568062542163, −2.27767983567120718258852681783,
0.24180487227083459192103064091, 1.56284211725174995451881222743, 2.42441246894867687522291625691, 3.68960547837397041636009258577, 5.65502845650550796114012482173, 6.04833805952108573399105828853, 7.44394832846832267683733722316, 8.373809881170775037067204240049, 9.097450948511925333605159400527, 10.26222897437973536190597119266