L(s) = 1 | + (−7.08 + 8.64i)5-s + (3.01 − 3.01i)7-s − 64.6i·11-s + (46.4 + 46.4i)13-s + (51.1 + 51.1i)17-s + 75.4i·19-s + (−131. + 131. i)23-s + (−24.6 − 122. i)25-s − 279.·29-s − 7.69·31-s + (4.72 + 47.5i)35-s + (−152. + 152. i)37-s + 295. i·41-s + (200. + 200. i)43-s + (−292. − 292. i)47-s + ⋯ |
L(s) = 1 | + (−0.633 + 0.773i)5-s + (0.163 − 0.163i)7-s − 1.77i·11-s + (0.992 + 0.992i)13-s + (0.730 + 0.730i)17-s + 0.911i·19-s + (−1.19 + 1.19i)23-s + (−0.196 − 0.980i)25-s − 1.78·29-s − 0.0445·31-s + (0.0228 + 0.229i)35-s + (−0.678 + 0.678i)37-s + 1.12i·41-s + (0.710 + 0.710i)43-s + (−0.907 − 0.907i)47-s + ⋯ |
Λ(s)=(=(360s/2ΓC(s)L(s)(−0.480−0.877i)Λ(4−s)
Λ(s)=(=(360s/2ΓC(s+3/2)L(s)(−0.480−0.877i)Λ(1−s)
Degree: |
2 |
Conductor: |
360
= 23⋅32⋅5
|
Sign: |
−0.480−0.877i
|
Analytic conductor: |
21.2406 |
Root analytic conductor: |
4.60876 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ360(233,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 360, ( :3/2), −0.480−0.877i)
|
Particular Values
L(2) |
≈ |
1.074087705 |
L(21) |
≈ |
1.074087705 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(7.08−8.64i)T |
good | 7 | 1+(−3.01+3.01i)T−343iT2 |
| 11 | 1+64.6iT−1.33e3T2 |
| 13 | 1+(−46.4−46.4i)T+2.19e3iT2 |
| 17 | 1+(−51.1−51.1i)T+4.91e3iT2 |
| 19 | 1−75.4iT−6.85e3T2 |
| 23 | 1+(131.−131.i)T−1.21e4iT2 |
| 29 | 1+279.T+2.43e4T2 |
| 31 | 1+7.69T+2.97e4T2 |
| 37 | 1+(152.−152.i)T−5.06e4iT2 |
| 41 | 1−295.iT−6.89e4T2 |
| 43 | 1+(−200.−200.i)T+7.95e4iT2 |
| 47 | 1+(292.+292.i)T+1.03e5iT2 |
| 53 | 1+(212.−212.i)T−1.48e5iT2 |
| 59 | 1+55.6T+2.05e5T2 |
| 61 | 1−19.2T+2.26e5T2 |
| 67 | 1+(−437.+437.i)T−3.00e5iT2 |
| 71 | 1−582.iT−3.57e5T2 |
| 73 | 1+(172.+172.i)T+3.89e5iT2 |
| 79 | 1−767.iT−4.93e5T2 |
| 83 | 1+(−369.+369.i)T−5.71e5iT2 |
| 89 | 1−394.T+7.04e5T2 |
| 97 | 1+(291.−291.i)T−9.12e5iT2 |
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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
−11.30280207091846633208830039004, −10.65134215951136505343434211013, −9.498144221336004029887619425035, −8.280671328786790633535521250052, −7.77145432253395873453379381751, −6.37895371652999454627907021396, −5.75594916142004425471827716725, −3.87805787783619885301149181654, −3.41099356271653098110444985021, −1.49656815164524061320103505552,
0.38436512624979855600143019174, 1.98529662302810052589802718871, 3.66928289639470738223035424319, 4.72101251344881296527629648789, 5.62542505747591778848919516700, 7.12714974057478889152223865027, 7.85896077318672755315086188889, 8.837728898586267877477508341284, 9.721664733791871697666499226180, 10.73806671334384533963576421934