L(s) = 1 | + (−2.34 + 1.58i)2-s + (2.99 − 7.41i)4-s + 6.32i·5-s + 10·7-s + (4.69 + 22.1i)8-s + (−10.0 − 14.8i)10-s + 37.9i·11-s + 59.3i·13-s + (−23.4 + 15.8i)14-s + (−46.0 − 44.4i)16-s − 75.0·17-s + 118. i·19-s + (46.9 + 18.9i)20-s + (−60.0 − 88.9i)22-s + 150.·23-s + ⋯ |
L(s) = 1 | + (−0.829 + 0.559i)2-s + (0.374 − 0.927i)4-s + 0.565i·5-s + 0.539·7-s + (0.207 + 0.978i)8-s + (−0.316 − 0.469i)10-s + 1.04i·11-s + 1.26i·13-s + (−0.447 + 0.301i)14-s + (−0.718 − 0.695i)16-s − 1.07·17-s + 1.43i·19-s + (0.524 + 0.212i)20-s + (−0.581 − 0.862i)22-s + 1.36·23-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(−0.207−0.978i)Λ(4−s)
Λ(s)=(=(72s/2ΓC(s+3/2)L(s)(−0.207−0.978i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
−0.207−0.978i
|
Analytic conductor: |
4.24813 |
Root analytic conductor: |
2.06110 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(37,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :3/2), −0.207−0.978i)
|
Particular Values
L(2) |
≈ |
0.592975+0.731787i |
L(21) |
≈ |
0.592975+0.731787i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(2.34−1.58i)T |
| 3 | 1 |
good | 5 | 1−6.32iT−125T2 |
| 7 | 1−10T+343T2 |
| 11 | 1−37.9iT−1.33e3T2 |
| 13 | 1−59.3iT−2.19e3T2 |
| 17 | 1+75.0T+4.91e3T2 |
| 19 | 1−118.iT−6.85e3T2 |
| 23 | 1−150.T+1.21e4T2 |
| 29 | 1+246.iT−2.43e4T2 |
| 31 | 1−62T+2.97e4T2 |
| 37 | 1−59.3iT−5.06e4T2 |
| 41 | 1+375.T+6.89e4T2 |
| 43 | 1+118.iT−7.95e4T2 |
| 47 | 1−450.T+1.03e5T2 |
| 53 | 1+132.iT−1.48e5T2 |
| 59 | 1−733.iT−2.05e5T2 |
| 61 | 1+533.iT−2.26e5T2 |
| 67 | 1+711.iT−3.00e5T2 |
| 71 | 1+3.57e5T2 |
| 73 | 1−30T+3.89e5T2 |
| 79 | 1−94T+4.93e5T2 |
| 83 | 1+670.iT−5.71e5T2 |
| 89 | 1−750.T+7.04e5T2 |
| 97 | 1−130T+9.12e5T2 |
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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
−14.75626615954404905656443483582, −13.72706490407965680655699960224, −11.93895753174482768947125277454, −10.91081478722974632289438432422, −9.829792816275799073305332603816, −8.683527573471398514215268319860, −7.36016460731941364759068163279, −6.43642249899004322837223289722, −4.66818566401842876303810128580, −1.95698788669787030287002228668,
0.843581455303045577154825550162, 2.98964440745171456406232368856, 4.98240964575716959785315717052, 6.96759718286122523807758828622, 8.437182450692687677778447787886, 9.006553114226092258440059070768, 10.68271395759717941866855520390, 11.26810327354987394760084865014, 12.69159344472675627151117507668, 13.43166472160846117522934532718