L(s) = 1 | − 1.73i·5-s + (−2.5 + 2.59i)13-s + (−1.5 + 2.59i)17-s + (3 + 1.73i)19-s + (3 + 5.19i)23-s + 2.00·25-s + (1.5 + 2.59i)29-s + 3.46i·31-s + (7.5 − 4.33i)37-s + (4.5 − 2.59i)41-s + (4 − 6.92i)43-s + 3.46i·47-s + (−3.5 − 6.06i)49-s + 3·53-s + (6 + 3.46i)59-s + ⋯ |
L(s) = 1 | − 0.774i·5-s + (−0.693 + 0.720i)13-s + (−0.363 + 0.630i)17-s + (0.688 + 0.397i)19-s + (0.625 + 1.08i)23-s + 0.400·25-s + (0.278 + 0.482i)29-s + 0.622i·31-s + (1.23 − 0.711i)37-s + (0.702 − 0.405i)41-s + (0.609 − 1.05i)43-s + 0.505i·47-s + (−0.5 − 0.866i)49-s + 0.412·53-s + (0.781 + 0.450i)59-s + ⋯ |
Λ(s)=(=(1872s/2ΓC(s)L(s)(0.964−0.265i)Λ(2−s)
Λ(s)=(=(1872s/2ΓC(s+1/2)L(s)(0.964−0.265i)Λ(1−s)
Degree: |
2 |
Conductor: |
1872
= 24⋅32⋅13
|
Sign: |
0.964−0.265i
|
Analytic conductor: |
14.9479 |
Root analytic conductor: |
3.86626 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1872(1297,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1872, ( :1/2), 0.964−0.265i)
|
Particular Values
L(1) |
≈ |
1.649925025 |
L(21) |
≈ |
1.649925025 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 13 | 1+(2.5−2.59i)T |
good | 5 | 1+1.73iT−5T2 |
| 7 | 1+(3.5+6.06i)T2 |
| 11 | 1+(5.5−9.52i)T2 |
| 17 | 1+(1.5−2.59i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−3−1.73i)T+(9.5+16.4i)T2 |
| 23 | 1+(−3−5.19i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−1.5−2.59i)T+(−14.5+25.1i)T2 |
| 31 | 1−3.46iT−31T2 |
| 37 | 1+(−7.5+4.33i)T+(18.5−32.0i)T2 |
| 41 | 1+(−4.5+2.59i)T+(20.5−35.5i)T2 |
| 43 | 1+(−4+6.92i)T+(−21.5−37.2i)T2 |
| 47 | 1−3.46iT−47T2 |
| 53 | 1−3T+53T2 |
| 59 | 1+(−6−3.46i)T+(29.5+51.0i)T2 |
| 61 | 1+(0.5−0.866i)T+(−30.5−52.8i)T2 |
| 67 | 1+(3−1.73i)T+(33.5−58.0i)T2 |
| 71 | 1+(−3−1.73i)T+(35.5+61.4i)T2 |
| 73 | 1+1.73iT−73T2 |
| 79 | 1+4T+79T2 |
| 83 | 1+13.8iT−83T2 |
| 89 | 1+(−6+3.46i)T+(44.5−77.0i)T2 |
| 97 | 1+(−6−3.46i)T+(48.5+84.0i)T2 |
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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
−9.123077605873459863605724253906, −8.697988548714246519582033478004, −7.61857570742759708878695674415, −7.04737512891881071243488383493, −5.96159616064971977788446866911, −5.16666675086398900658768968531, −4.41897364129475346952967322849, −3.45064800798618838084577413872, −2.18501010135135955767249891651, −1.04391732407829540658714043117,
0.75508942227275846024388682668, 2.58162833387306437816196125022, 2.94410895101607160802272327130, 4.33594373541198768671724370219, 5.07917652869731294007720257944, 6.12922487867576987838812897419, 6.85481181379165610606297371129, 7.57821239145530444057098171169, 8.299138643179058367045747390571, 9.393889983743260479636102295743