L(s) = 1 | − 2.23i·3-s − 3i·5-s + 2.23·7-s − 2.00·9-s − 4.47i·11-s − i·13-s − 6.70·15-s − 3·17-s + 4.47i·19-s − 5.00i·21-s − 8.94·23-s − 4·25-s − 2.23i·27-s + 10i·29-s − 10.0·33-s + ⋯ |
L(s) = 1 | − 1.29i·3-s − 1.34i·5-s + 0.845·7-s − 0.666·9-s − 1.34i·11-s − 0.277i·13-s − 1.73·15-s − 0.727·17-s + 1.02i·19-s − 1.09i·21-s − 1.86·23-s − 0.800·25-s − 0.430i·27-s + 1.85i·29-s − 1.74·33-s + ⋯ |
Λ(s)=(=(3328s/2ΓC(s)L(s)(−0.707−0.707i)Λ(2−s)
Λ(s)=(=(3328s/2ΓC(s+1/2)L(s)(−0.707−0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
3328
= 28⋅13
|
Sign: |
−0.707−0.707i
|
Analytic conductor: |
26.5742 |
Root analytic conductor: |
5.15501 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3328(1665,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3328, ( :1/2), −0.707−0.707i)
|
Particular Values
L(1) |
≈ |
1.265525316 |
L(21) |
≈ |
1.265525316 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+iT |
good | 3 | 1+2.23iT−3T2 |
| 5 | 1+3iT−5T2 |
| 7 | 1−2.23T+7T2 |
| 11 | 1+4.47iT−11T2 |
| 17 | 1+3T+17T2 |
| 19 | 1−4.47iT−19T2 |
| 23 | 1+8.94T+23T2 |
| 29 | 1−10iT−29T2 |
| 31 | 1+31T2 |
| 37 | 1+3iT−37T2 |
| 41 | 1+41T2 |
| 43 | 1−6.70iT−43T2 |
| 47 | 1+2.23T+47T2 |
| 53 | 1+4iT−53T2 |
| 59 | 1+4.47iT−59T2 |
| 61 | 1−61T2 |
| 67 | 1+13.4iT−67T2 |
| 71 | 1−6.70T+71T2 |
| 73 | 1+14T+73T2 |
| 79 | 1−8.94T+79T2 |
| 83 | 1+17.8iT−83T2 |
| 89 | 1−10T+89T2 |
| 97 | 1+2T+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
−8.068793140220475787863403631665, −7.78171600246497402485886233207, −6.58729584588435536777439207868, −5.94253520243105416126131206293, −5.23116982894672747637051606040, −4.39125785254593678461574087875, −3.35588060359060981566985045467, −1.92462084059525295215857969172, −1.41432921684726186387629782177, −0.36781695494391838827899421765,
2.00428988594600100214551510481, 2.63571475388658294722741457533, 3.93073584593977016905232276533, 4.31850489629127107782029398352, 5.05303213584734202699689688518, 6.10746565822432930091051026768, 6.90046795087838769933835325198, 7.54383883445810561823918469716, 8.383669195273141069950298744852, 9.329269829423901381406771024068