L(s) = 1 | − 2.87i·2-s + (−2.93 − 0.599i)3-s − 4.24·4-s + (−6.53 + 3.77i)5-s + (−1.72 + 8.44i)6-s + (2.05 − 6.69i)7-s + 0.709i·8-s + (8.28 + 3.52i)9-s + (10.8 + 18.7i)10-s + (−13.1 − 7.59i)11-s + (12.4 + 2.54i)12-s + (4.30 − 7.45i)13-s + (−19.2 − 5.90i)14-s + (21.4 − 7.17i)15-s − 14.9·16-s + (−4.60 + 2.66i)17-s + ⋯ |
L(s) = 1 | − 1.43i·2-s + (−0.979 − 0.199i)3-s − 1.06·4-s + (−1.30 + 0.754i)5-s + (−0.286 + 1.40i)6-s + (0.293 − 0.955i)7-s + 0.0886i·8-s + (0.920 + 0.391i)9-s + (1.08 + 1.87i)10-s + (−1.19 − 0.690i)11-s + (1.04 + 0.212i)12-s + (0.331 − 0.573i)13-s + (−1.37 − 0.421i)14-s + (1.43 − 0.478i)15-s − 0.934·16-s + (−0.271 + 0.156i)17-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(−0.894−0.446i)Λ(3−s)
Λ(s)=(=(63s/2ΓC(s+1)L(s)(−0.894−0.446i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
−0.894−0.446i
|
Analytic conductor: |
1.71662 |
Root analytic conductor: |
1.31020 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ63(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 63, ( :1), −0.894−0.446i)
|
Particular Values
L(23) |
≈ |
0.114141+0.484698i |
L(21) |
≈ |
0.114141+0.484698i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(2.93+0.599i)T |
| 7 | 1+(−2.05+6.69i)T |
good | 2 | 1+2.87iT−4T2 |
| 5 | 1+(6.53−3.77i)T+(12.5−21.6i)T2 |
| 11 | 1+(13.1+7.59i)T+(60.5+104.i)T2 |
| 13 | 1+(−4.30+7.45i)T+(−84.5−146.i)T2 |
| 17 | 1+(4.60−2.66i)T+(144.5−250.i)T2 |
| 19 | 1+(0.417−0.722i)T+(−180.5−312.i)T2 |
| 23 | 1+(−33.8+19.5i)T+(264.5−458.i)T2 |
| 29 | 1+(−12.5+7.25i)T+(420.5−728.i)T2 |
| 31 | 1−12.7T+961T2 |
| 37 | 1+(11.7−20.3i)T+(−684.5−1.18e3i)T2 |
| 41 | 1+(13.4+7.75i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(−0.448−0.776i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+2.35iT−2.20e3T2 |
| 53 | 1+(31.4−18.1i)T+(1.40e3−2.43e3i)T2 |
| 59 | 1+48.6iT−3.48e3T2 |
| 61 | 1+29.6T+3.72e3T2 |
| 67 | 1−92.5T+4.48e3T2 |
| 71 | 1−15.1iT−5.04e3T2 |
| 73 | 1+(46.8+81.0i)T+(−2.66e3+4.61e3i)T2 |
| 79 | 1+82.1T+6.24e3T2 |
| 83 | 1+(−127.+73.3i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1+(−92.3−53.2i)T+(3.96e3+6.85e3i)T2 |
| 97 | 1+(26.0+45.0i)T+(−4.70e3+8.14e3i)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
−13.54404591655057303220448707059, −12.64775426213615962049255522311, −11.48973538947100087706282722076, −10.81727664390609053434988370964, −10.41371990197610202879685720993, −8.049431440468640334504178840223, −6.83515382450712050301286010330, −4.63549392031665671306821864464, −3.22424197911385682344932574576, −0.52148987682429679346737572832,
4.64817414435236099893798663826, 5.37419393783963400787342576645, 6.94971681062098608054258948714, 8.010580484355659562348967519937, 9.129382956516028007732193436454, 11.10915189497396338730719081552, 12.01594526519611641049095567181, 13.08049999204700168644113594754, 14.96606081085873893862400389288, 15.69542239603633568335693625305