L(s) = 1 | + 3i·3-s + (−10 − 5i)5-s − 10i·7-s − 9·9-s + 46·11-s + 34i·13-s + (15 − 30i)15-s + 66i·17-s + 104·19-s + 30·21-s + 164i·23-s + (75 + 100i)25-s − 27i·27-s − 224·29-s + 72·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.539i·7-s − 0.333·9-s + 1.26·11-s + 0.725i·13-s + (0.258 − 0.516i)15-s + 0.941i·17-s + 1.25·19-s + 0.311·21-s + 1.48i·23-s + (0.599 + 0.800i)25-s − 0.192i·27-s − 1.43·29-s + 0.417·31-s + ⋯ |
Λ(s)=(=(240s/2ΓC(s)L(s)(0.447−0.894i)Λ(4−s)
Λ(s)=(=(240s/2ΓC(s+3/2)L(s)(0.447−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
240
= 24⋅3⋅5
|
Sign: |
0.447−0.894i
|
Analytic conductor: |
14.1604 |
Root analytic conductor: |
3.76303 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ240(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 240, ( :3/2), 0.447−0.894i)
|
Particular Values
L(2) |
≈ |
1.23325+0.762193i |
L(21) |
≈ |
1.23325+0.762193i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−3iT |
| 5 | 1+(10+5i)T |
good | 7 | 1+10iT−343T2 |
| 11 | 1−46T+1.33e3T2 |
| 13 | 1−34iT−2.19e3T2 |
| 17 | 1−66iT−4.91e3T2 |
| 19 | 1−104T+6.85e3T2 |
| 23 | 1−164iT−1.21e4T2 |
| 29 | 1+224T+2.43e4T2 |
| 31 | 1−72T+2.97e4T2 |
| 37 | 1+22iT−5.06e4T2 |
| 41 | 1−194T+6.89e4T2 |
| 43 | 1−108iT−7.95e4T2 |
| 47 | 1−480iT−1.03e5T2 |
| 53 | 1+286iT−1.48e5T2 |
| 59 | 1−426T+2.05e5T2 |
| 61 | 1−698T+2.26e5T2 |
| 67 | 1+328iT−3.00e5T2 |
| 71 | 1+188T+3.57e5T2 |
| 73 | 1−740iT−3.89e5T2 |
| 79 | 1−1.16e3T+4.93e5T2 |
| 83 | 1−412iT−5.71e5T2 |
| 89 | 1+1.20e3T+7.04e5T2 |
| 97 | 1+1.38e3iT−9.12e5T2 |
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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
−11.56443050906327530517220662032, −11.21397825538864847938256684229, −9.727043493866489091855245491148, −9.104298777559835004724024263880, −7.922391735247120713232863085489, −6.96437411932091214153296068620, −5.54211626932459469915570398594, −4.17196759682810013289975334958, −3.62648118495801271238709323632, −1.25387175509316289668548395249,
0.70148799396917029260058376361, 2.62346782158293543362061476002, 3.84986752814880570112904772219, 5.38159035300773826298257328281, 6.65502269083394473260254694649, 7.44222762645074758291071728648, 8.475669382080289593855659949869, 9.446368007447710611995930170548, 10.77716432500795205941830858081, 11.82743919665936893472680894648