L(s) = 1 | − i·3-s − i·5-s + 3·7-s + 2·9-s + 2i·11-s + i·13-s − 15-s − 3·17-s − 2i·19-s − 3i·21-s + 4·23-s + 4·25-s − 5i·27-s + 2i·29-s − 4·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.13·7-s + 0.666·9-s + 0.603i·11-s + 0.277i·13-s − 0.258·15-s − 0.727·17-s − 0.458i·19-s − 0.654i·21-s + 0.834·23-s + 0.800·25-s − 0.962i·27-s + 0.371i·29-s − 0.718·31-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) |
≈ |
2.400789823 |
L(21) |
≈ |
2.400789823 |
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+iT−3T2 |
| 5 | 1+iT−5T2 |
| 7 | 1−3T+7T2 |
| 11 | 1−2iT−11T2 |
| 17 | 1+3T+17T2 |
| 19 | 1+2iT−19T2 |
| 23 | 1−4T+23T2 |
| 29 | 1−2iT−29T2 |
| 31 | 1+4T+31T2 |
| 37 | 1+5iT−37T2 |
| 41 | 1−12T+41T2 |
| 43 | 1−7iT−43T2 |
| 47 | 1−9T+47T2 |
| 53 | 1+4iT−53T2 |
| 59 | 1−6iT−59T2 |
| 61 | 1+4iT−61T2 |
| 67 | 1−10iT−67T2 |
| 71 | 1+15T+71T2 |
| 73 | 1−2T+73T2 |
| 79 | 1−8T+79T2 |
| 83 | 1−4iT−83T2 |
| 89 | 1+2T+89T2 |
| 97 | 1−10T+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
−8.628387802360297444039085334453, −7.50211611712680101994387131035, −7.32767881993696853970978007488, −6.42416732087092013598387706853, −5.38607867962064949987032330061, −4.60549786053395597259857643766, −4.17543957128242856597877921997, −2.63266241948440037720055765942, −1.76602891595524199945264321345, −0.944510207715478198284809024647,
1.04747734778876854886060885548, 2.19254625144669468143236401663, 3.24796225218268772921656850650, 4.15457340430832945190843238100, 4.80604899610561460037535656113, 5.56761019982169990166111603987, 6.49476624921280522772804785712, 7.35400998102095319157005823962, 7.905114463359413762411237583990, 8.880157672028371211683869038403