L(s) = 1 | + (−4.5 − 7.79i)5-s + (−15.5 + 26.8i)7-s + (7.5 − 12.9i)11-s + (18.5 + 32.0i)13-s + 42·17-s + 28·19-s + (−97.5 − 168. i)23-s + (22 − 38.1i)25-s + (55.5 − 96.1i)29-s + (−102.5 − 177. i)31-s + 279·35-s − 166·37-s + (−130.5 − 226. i)41-s + (−21.5 + 37.2i)43-s + (−88.5 + 153. i)47-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.697i)5-s + (−0.836 + 1.44i)7-s + (0.205 − 0.356i)11-s + (0.394 + 0.683i)13-s + 0.599·17-s + 0.338·19-s + (−0.883 − 1.53i)23-s + (0.175 − 0.304i)25-s + (0.355 − 0.615i)29-s + (−0.593 − 1.02i)31-s + 1.34·35-s − 0.737·37-s + (−0.497 − 0.860i)41-s + (−0.0762 + 0.132i)43-s + (−0.274 + 0.475i)47-s + ⋯ |
Λ(s)=(=(432s/2ΓC(s)L(s)(−0.173+0.984i)Λ(4−s)
Λ(s)=(=(432s/2ΓC(s+3/2)L(s)(−0.173+0.984i)Λ(1−s)
Degree: |
2 |
Conductor: |
432
= 24⋅33
|
Sign: |
−0.173+0.984i
|
Analytic conductor: |
25.4888 |
Root analytic conductor: |
5.04864 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ432(289,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 432, ( :3/2), −0.173+0.984i)
|
Particular Values
L(2) |
≈ |
0.9414357461 |
L(21) |
≈ |
0.9414357461 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1+(4.5+7.79i)T+(−62.5+108.i)T2 |
| 7 | 1+(15.5−26.8i)T+(−171.5−297.i)T2 |
| 11 | 1+(−7.5+12.9i)T+(−665.5−1.15e3i)T2 |
| 13 | 1+(−18.5−32.0i)T+(−1.09e3+1.90e3i)T2 |
| 17 | 1−42T+4.91e3T2 |
| 19 | 1−28T+6.85e3T2 |
| 23 | 1+(97.5+168.i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(−55.5+96.1i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+(102.5+177.i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+166T+5.06e4T2 |
| 41 | 1+(130.5+226.i)T+(−3.44e4+5.96e4i)T2 |
| 43 | 1+(21.5−37.2i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+(88.5−153.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1+114T+1.48e5T2 |
| 59 | 1+(79.5+137.i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(95.5−165.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(210.5+364.i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1−156T+3.57e5T2 |
| 73 | 1−182T+3.89e5T2 |
| 79 | 1+(−566.5+981.i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1+(−541.5+937.i)T+(−2.85e5−4.95e5i)T2 |
| 89 | 1−1.05e3T+7.04e5T2 |
| 97 | 1+(−450.5+780.i)T+(−4.56e5−7.90e5i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.40894919483553110190630612972, −9.343681836236453533628441478390, −8.774372866397727631277820961578, −7.965799518611042591148609586694, −6.48104280476927233795848594527, −5.83381891554021431946591178676, −4.63672822057916849006697735645, −3.42685737800402894206051423204, −2.13365243459226088686900512136, −0.33556561062097868471125577813,
1.23253564219819021721158489995, 3.31349511894806453452290979204, 3.69613215484793512993523963489, 5.24115256974848884317836032062, 6.55978718274516047111938178397, 7.24356115688758323629364233179, 7.969441525830167702290528977839, 9.410260275127040418797247643904, 10.23148323017230714257345455513, 10.77917725759148092271318369268