L(s) = 1 | − 1.80i·2-s − 0.561i·3-s − 1.25·4-s + (−0.178 − 2.22i)5-s − 1.01·6-s + 3.11i·7-s − 1.34i·8-s + 2.68·9-s + (−4.01 + 0.321i)10-s − 5.61·11-s + 0.703i·12-s + 1.24i·13-s + 5.61·14-s + (−1.25 + 0.100i)15-s − 4.93·16-s + ⋯ |
L(s) = 1 | − 1.27i·2-s − 0.324i·3-s − 0.626·4-s + (−0.0796 − 0.996i)5-s − 0.413·6-s + 1.17i·7-s − 0.476i·8-s + 0.894·9-s + (−1.27 + 0.101i)10-s − 1.69·11-s + 0.203i·12-s + 0.346i·13-s + 1.49·14-s + (−0.323 + 0.0258i)15-s − 1.23·16-s + ⋯ |
Λ(s)=(=(1445s/2ΓC(s)L(s)(−0.0796−0.996i)Λ(2−s)
Λ(s)=(=(1445s/2ΓC(s+1/2)L(s)(−0.0796−0.996i)Λ(1−s)
Degree: |
2 |
Conductor: |
1445
= 5⋅172
|
Sign: |
−0.0796−0.996i
|
Analytic conductor: |
11.5383 |
Root analytic conductor: |
3.39681 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1445(579,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1445, ( :1/2), −0.0796−0.996i)
|
Particular Values
L(1) |
≈ |
0.5281995747 |
L(21) |
≈ |
0.5281995747 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(0.178+2.22i)T |
| 17 | 1 |
good | 2 | 1+1.80iT−2T2 |
| 3 | 1+0.561iT−3T2 |
| 7 | 1−3.11iT−7T2 |
| 11 | 1+5.61T+11T2 |
| 13 | 1−1.24iT−13T2 |
| 19 | 1+4T+19T2 |
| 23 | 1+2.32iT−23T2 |
| 29 | 1+6.62T+29T2 |
| 31 | 1+4.55T+31T2 |
| 37 | 1−1.90iT−37T2 |
| 41 | 1+5.92T+41T2 |
| 43 | 1−2.04iT−43T2 |
| 47 | 1+4.85iT−47T2 |
| 53 | 1+9.11iT−53T2 |
| 59 | 1+6T+59T2 |
| 61 | 1+5.65T+61T2 |
| 67 | 1+8.46iT−67T2 |
| 71 | 1−8.79T+71T2 |
| 73 | 1−1.56iT−73T2 |
| 79 | 1−4.91T+79T2 |
| 83 | 1−3.94iT−83T2 |
| 89 | 1−10.6T+89T2 |
| 97 | 1+12.3iT−97T2 |
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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
−9.107423508847395826202542639438, −8.349077703114772634896844755732, −7.51509459494567180323475018110, −6.40102770068463814110379978601, −5.31506757686974333587922650625, −4.62450926391204792615116985542, −3.53701577490347268637141418718, −2.26278586586650891631245636533, −1.81163493190699557251385266472, −0.19059573793131521051834249299,
2.13380630298043453059573986944, 3.42702900355250888737398816116, 4.39174751880946961855759722451, 5.30969205085939295447547132646, 6.16924730688407643678359690870, 7.14496395030185312949922233893, 7.48101120322767627993840408366, 8.011886429356934704579080719973, 9.256694185231227719910744606296, 10.29127679603532216063845363353