L(s) = 1 | + (−4.67 − 1.77i)5-s − 9.67i·7-s + 8.58i·11-s − 19.0i·13-s − 26.2·17-s − 26.6·19-s − 25.8·23-s + (18.7 + 16.5i)25-s + 18.3i·29-s + 7.56·31-s + (−17.1 + 45.2i)35-s + 47.0i·37-s − 0.602i·41-s − 62.2i·43-s + 79.2·47-s + ⋯ |
L(s) = 1 | + (−0.934 − 0.354i)5-s − 1.38i·7-s + 0.780i·11-s − 1.46i·13-s − 1.54·17-s − 1.40·19-s − 1.12·23-s + (0.748 + 0.663i)25-s + 0.633i·29-s + 0.243·31-s + (−0.490 + 1.29i)35-s + 1.27i·37-s − 0.0146i·41-s − 1.44i·43-s + 1.68·47-s + ⋯ |
Λ(s)=(=(2160s/2ΓC(s)L(s)(0.354−0.934i)Λ(3−s)
Λ(s)=(=(2160s/2ΓC(s+1)L(s)(0.354−0.934i)Λ(1−s)
Degree: |
2 |
Conductor: |
2160
= 24⋅33⋅5
|
Sign: |
0.354−0.934i
|
Analytic conductor: |
58.8557 |
Root analytic conductor: |
7.67174 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2160(1889,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2160, ( :1), 0.354−0.934i)
|
Particular Values
L(23) |
≈ |
0.3693540965 |
L(21) |
≈ |
0.3693540965 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(4.67+1.77i)T |
good | 7 | 1+9.67iT−49T2 |
| 11 | 1−8.58iT−121T2 |
| 13 | 1+19.0iT−169T2 |
| 17 | 1+26.2T+289T2 |
| 19 | 1+26.6T+361T2 |
| 23 | 1+25.8T+529T2 |
| 29 | 1−18.3iT−841T2 |
| 31 | 1−7.56T+961T2 |
| 37 | 1−47.0iT−1.36e3T2 |
| 41 | 1+0.602iT−1.68e3T2 |
| 43 | 1+62.2iT−1.84e3T2 |
| 47 | 1−79.2T+2.20e3T2 |
| 53 | 1−80.9T+2.80e3T2 |
| 59 | 1−15.4iT−3.48e3T2 |
| 61 | 1+62.7T+3.72e3T2 |
| 67 | 1+48.3iT−4.48e3T2 |
| 71 | 1+99.6iT−5.04e3T2 |
| 73 | 1−60.7iT−5.32e3T2 |
| 79 | 1+60.8T+6.24e3T2 |
| 83 | 1+77.0T+6.88e3T2 |
| 89 | 1+28.7iT−7.92e3T2 |
| 97 | 1−23.3iT−9.40e3T2 |
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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
−8.767481431806117153547027984923, −8.308186673216523821059431759955, −7.38112493236926714436762020224, −7.01265634261423549356667470916, −5.94452342456817243759195605395, −4.67507946075053921918137043778, −4.28625720462524095977880652932, −3.46327291747985199015098707775, −2.15377653019875470319429867648, −0.75190572412762293496804776055,
0.12566035570467189255189909390, 2.06788284377989074428568083138, 2.64761915605545553396169740500, 4.08108478622502857701766072504, 4.37565980028403929525772135351, 5.80027520943528213257954351681, 6.34399751351649491437923738059, 7.11485056684166560939632309963, 8.183976217269618576808322692661, 8.776958457061252613987336435474