L(s) = 1 | − i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (−1 + 2i)15-s + 2i·17-s − 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·31-s + 2i·33-s + (−2 + 4i)35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.43·31-s + 0.348i·33-s + (−0.338 + 0.676i)35-s + ⋯ |
Λ(s)=(=(240s/2ΓC(s)L(s)(−0.447+0.894i)Λ(2−s)
Λ(s)=(=(240s/2ΓC(s+1/2)L(s)(−0.447+0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
240
= 24⋅3⋅5
|
Sign: |
−0.447+0.894i
|
Analytic conductor: |
1.91640 |
Root analytic conductor: |
1.38434 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ240(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 240, ( :1/2), −0.447+0.894i)
|
Particular Values
L(1) |
≈ |
0.465077−0.752510i |
L(21) |
≈ |
0.465077−0.752510i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+iT |
| 5 | 1+(2+i)T |
good | 7 | 1+2iT−7T2 |
| 11 | 1+2T+11T2 |
| 13 | 1+6iT−13T2 |
| 17 | 1−2iT−17T2 |
| 19 | 1+19T2 |
| 23 | 1+4iT−23T2 |
| 29 | 1+29T2 |
| 31 | 1−8T+31T2 |
| 37 | 1−2iT−37T2 |
| 41 | 1−2T+41T2 |
| 43 | 1+4iT−43T2 |
| 47 | 1−8iT−47T2 |
| 53 | 1+6iT−53T2 |
| 59 | 1−10T+59T2 |
| 61 | 1−2T+61T2 |
| 67 | 1−8iT−67T2 |
| 71 | 1+12T+71T2 |
| 73 | 1−4iT−73T2 |
| 79 | 1+79T2 |
| 83 | 1+4iT−83T2 |
| 89 | 1−10T+89T2 |
| 97 | 1+8iT−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
−11.97204191946259444215266705660, −10.85038935669694538881413320876, −10.13056002929891195176822211936, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.00717100446239630133784520713, −5.61924101980997412333828064393, −4.36151973355833603134652685788, −2.98301548675506377793302167956, −0.72864190327182967399520039641,
2.58342719920451962723501294913, 3.94695391081871525948763714951, 5.02781841842898122069153631094, 6.41014651359295177422336904223, 7.53127668903399757597493950167, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.60982430904786714834563426049, 11.65746295943836746628078584697, 11.97373279112430878810242545265