L(s) = 1 | + (5.65 + 0.276i)2-s + (−8.82 + 12.8i)3-s + (31.8 + 3.12i)4-s + 100. i·5-s + (−53.4 + 70.1i)6-s − 49i·7-s + (179. + 26.4i)8-s + (−87.2 − 226. i)9-s + (−27.6 + 565. i)10-s − 5.60·11-s + (−321. + 381. i)12-s − 1.04e3·13-s + (13.5 − 276. i)14-s + (−1.28e3 − 883. i)15-s + (1.00e3 + 198. i)16-s + 333. i·17-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0488i)2-s + (−0.566 + 0.824i)3-s + (0.995 + 0.0975i)4-s + 1.79i·5-s + (−0.605 + 0.795i)6-s − 0.377i·7-s + (0.989 + 0.146i)8-s + (−0.358 − 0.933i)9-s + (−0.0874 + 1.78i)10-s − 0.0139·11-s + (−0.643 + 0.765i)12-s − 1.71·13-s + (0.0184 − 0.377i)14-s + (−1.47 − 1.01i)15-s + (0.980 + 0.194i)16-s + 0.279i·17-s + ⋯ |
Λ(s)=(=(84s/2ΓC(s)L(s)(−0.765−0.643i)Λ(6−s)
Λ(s)=(=(84s/2ΓC(s+5/2)L(s)(−0.765−0.643i)Λ(1−s)
Degree: |
2 |
Conductor: |
84
= 22⋅3⋅7
|
Sign: |
−0.765−0.643i
|
Analytic conductor: |
13.4722 |
Root analytic conductor: |
3.67045 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ84(71,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 84, ( :5/2), −0.765−0.643i)
|
Particular Values
L(3) |
≈ |
0.811127+2.22348i |
L(21) |
≈ |
0.811127+2.22348i |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−5.65−0.276i)T |
| 3 | 1+(8.82−12.8i)T |
| 7 | 1+49iT |
good | 5 | 1−100.iT−3.12e3T2 |
| 11 | 1+5.60T+1.61e5T2 |
| 13 | 1+1.04e3T+3.71e5T2 |
| 17 | 1−333.iT−1.41e6T2 |
| 19 | 1−1.63e3iT−2.47e6T2 |
| 23 | 1−3.80e3T+6.43e6T2 |
| 29 | 1−3.37e3iT−2.05e7T2 |
| 31 | 1+2.26e3iT−2.86e7T2 |
| 37 | 1−2.88e3T+6.93e7T2 |
| 41 | 1−680.iT−1.15e8T2 |
| 43 | 1+1.09e4iT−1.47e8T2 |
| 47 | 1+3.23e3T+2.29e8T2 |
| 53 | 1−3.16e4iT−4.18e8T2 |
| 59 | 1−2.68e4T+7.14e8T2 |
| 61 | 1−2.47e4T+8.44e8T2 |
| 67 | 1−3.14e4iT−1.35e9T2 |
| 71 | 1−3.66e4T+1.80e9T2 |
| 73 | 1−6.91e4T+2.07e9T2 |
| 79 | 1−2.56e4iT−3.07e9T2 |
| 83 | 1−5.31e4T+3.93e9T2 |
| 89 | 1+7.83e4iT−5.58e9T2 |
| 97 | 1+8.62e4T+8.58e9T2 |
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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
−14.13476023423403372380300171587, −12.51932117857791636037411104588, −11.44460037548220828425813433136, −10.62781214387047819660444326845, −9.917501723248082571595986378083, −7.41346617852586734671390349794, −6.59102999386025911627533183282, −5.29323329409033052942833429554, −3.84493678493584150235908436746, −2.70511662558157729263095340783,
0.74953769158305675442226443545, 2.32050001275252202408379434117, 4.85605152463149893673805974437, 5.22034295946761928450006174397, 6.82561116903121002234755841981, 8.040388689160035710476242156758, 9.516382776297569531509861885921, 11.32360292601377780723470697703, 12.17035102400941542773168846627, 12.80791590068433101417555658404