L(s) = 1 | + (−0.5 + 0.866i)3-s + 4.14·5-s + (−1.20 − 2.08i)7-s + (−0.499 − 0.866i)9-s + (1.73 − 3i)11-s + (−2.07 + 3.58i)15-s + (2.58 + 4.48i)17-s + (−1.73 − 3i)19-s + 2.41·21-s + (−1 + 1.73i)23-s + 12.1·25-s + 0.999·27-s + (−1.58 + 2.75i)29-s + 1.05·31-s + (1.73 + 3i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + 1.85·5-s + (−0.455 − 0.789i)7-s + (−0.166 − 0.288i)9-s + (0.522 − 0.904i)11-s + (−0.535 + 0.926i)15-s + (0.628 + 1.08i)17-s + (−0.397 − 0.688i)19-s + 0.526·21-s + (−0.208 + 0.361i)23-s + 2.43·25-s + 0.192·27-s + (−0.295 + 0.511i)29-s + 0.188·31-s + (0.301 + 0.522i)33-s + ⋯ |
Λ(s)=(=(2028s/2ΓC(s)L(s)(0.945+0.326i)Λ(2−s)
Λ(s)=(=(2028s/2ΓC(s+1/2)L(s)(0.945+0.326i)Λ(1−s)
Degree: |
2 |
Conductor: |
2028
= 22⋅3⋅132
|
Sign: |
0.945+0.326i
|
Analytic conductor: |
16.1936 |
Root analytic conductor: |
4.02413 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2028(529,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2028, ( :1/2), 0.945+0.326i)
|
Particular Values
L(1) |
≈ |
2.229198472 |
L(21) |
≈ |
2.229198472 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(0.5−0.866i)T |
| 13 | 1 |
good | 5 | 1−4.14T+5T2 |
| 7 | 1+(1.20+2.08i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−1.73+3i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−2.58−4.48i)T+(−8.5+14.7i)T2 |
| 19 | 1+(1.73+3i)T+(−9.5+16.4i)T2 |
| 23 | 1+(1−1.73i)T+(−11.5−19.9i)T2 |
| 29 | 1+(1.58−2.75i)T+(−14.5−25.1i)T2 |
| 31 | 1−1.05T+31T2 |
| 37 | 1+(−3.80+6.58i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−0.340+0.589i)T+(−20.5−35.5i)T2 |
| 43 | 1+(6.08+10.5i)T+(−21.5+37.2i)T2 |
| 47 | 1−10.3T+47T2 |
| 53 | 1−1.17T+53T2 |
| 59 | 1+(−5.87−10.1i)T+(−29.5+51.0i)T2 |
| 61 | 1+(2.5+4.33i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−1.20+2.08i)T+(−33.5−58.0i)T2 |
| 71 | 1+(1.73+3i)T+(−35.5+61.4i)T2 |
| 73 | 1+14.8T+73T2 |
| 79 | 1−1.82T+79T2 |
| 83 | 1+1.36T+83T2 |
| 89 | 1+(−3.46+6i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−8.81−15.2i)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
−9.056043901812262887443155033021, −8.785685681373790416726773589578, −7.34827461705914988263577064739, −6.49197604262756310956098737221, −5.88653772551027016389549044951, −5.35605791333955760175824432625, −4.13395860372219760280279058959, −3.30702900675108618496780519866, −2.09547899425946937071388341247, −0.917225692207709623533283917061,
1.28703322059709379227229484600, 2.20809680650128159447433353229, 2.90996562213687026031525663194, 4.55647441287209308633439098334, 5.42523657276802890413332171905, 6.09475373103717147297282332110, 6.54950712232408697235156165199, 7.47106461984342878078020704486, 8.566040852994502541309657990025, 9.396660362348267953784664868436