L(s) = 1 | + 1.61i·2-s + (−0.618 − 1.61i)3-s − 0.618·4-s + 2.61·5-s + (2.61 − 1.00i)6-s + (2.61 − 0.381i)7-s + 2.23i·8-s + (−2.23 + 2.00i)9-s + 4.23i·10-s − 2.23i·11-s + (0.381 + 1.00i)12-s − 6.85i·13-s + (0.618 + 4.23i)14-s + (−1.61 − 4.23i)15-s − 4.85·16-s + 1.47·17-s + ⋯ |
L(s) = 1 | + 1.14i·2-s + (−0.356 − 0.934i)3-s − 0.309·4-s + 1.17·5-s + (1.06 − 0.408i)6-s + (0.989 − 0.144i)7-s + 0.790i·8-s + (−0.745 + 0.666i)9-s + 1.33i·10-s − 0.674i·11-s + (0.110 + 0.288i)12-s − 1.90i·13-s + (0.165 + 1.13i)14-s + (−0.417 − 1.09i)15-s − 1.21·16-s + 0.357·17-s + ⋯ |
Λ(s)=(=(483s/2ΓC(s)L(s)(0.872−0.487i)Λ(2−s)
Λ(s)=(=(483s/2ΓC(s+1/2)L(s)(0.872−0.487i)Λ(1−s)
Degree: |
2 |
Conductor: |
483
= 3⋅7⋅23
|
Sign: |
0.872−0.487i
|
Analytic conductor: |
3.85677 |
Root analytic conductor: |
1.96386 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ483(461,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 483, ( :1/2), 0.872−0.487i)
|
Particular Values
L(1) |
≈ |
1.70537+0.444310i |
L(21) |
≈ |
1.70537+0.444310i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.618+1.61i)T |
| 7 | 1+(−2.61+0.381i)T |
| 23 | 1+iT |
good | 2 | 1−1.61iT−2T2 |
| 5 | 1−2.61T+5T2 |
| 11 | 1+2.23iT−11T2 |
| 13 | 1+6.85iT−13T2 |
| 17 | 1−1.47T+17T2 |
| 19 | 1−5.47iT−19T2 |
| 29 | 1−1.76iT−29T2 |
| 31 | 1−0.527iT−31T2 |
| 37 | 1−5.76T+37T2 |
| 41 | 1−0.236T+41T2 |
| 43 | 1−7.61T+43T2 |
| 47 | 1+8T+47T2 |
| 53 | 1−13.5iT−53T2 |
| 59 | 1+7.56T+59T2 |
| 61 | 1−0.854iT−61T2 |
| 67 | 1+10.6T+67T2 |
| 71 | 1+5.32iT−71T2 |
| 73 | 1+8.23iT−73T2 |
| 79 | 1+7.76T+79T2 |
| 83 | 1+10.7T+83T2 |
| 89 | 1−8.09T+89T2 |
| 97 | 1+1.94iT−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
−10.93903319445090684067843995684, −10.35366135591398325780194589712, −8.832238057650210038898710223558, −7.86770920276614059470928369557, −7.62623249791977015386424505931, −6.06400760783670640779205496331, −5.88950177272823048780742314086, −5.04290162688712199285765151897, −2.74318620388158491255135281557, −1.39647026846428147050162044608,
1.63055850491832581965360534497, 2.57099904858599971441612436658, 4.17227825748126203774238747866, 4.84315322848213187018152379581, 6.07842113778678663209393091182, 7.07449734467936911617856714257, 8.813734204348968379490012841369, 9.572211962557621922894317229963, 9.952290143113457973230213222877, 11.17991332144646222900145771611