L(s) = 1 | + 3-s + (−0.311 + 2.21i)5-s + (1.96 + 1.96i)7-s + 9-s + (0.870 − 0.870i)11-s + 5.88i·13-s + (−0.311 + 2.21i)15-s + (2.69 + 2.69i)17-s + (−2.40 + 2.40i)19-s + (1.96 + 1.96i)21-s + (−2.63 + 2.63i)23-s + (−4.80 − 1.38i)25-s + 27-s + (−7.43 − 7.43i)29-s − 7.72i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.139 + 0.990i)5-s + (0.743 + 0.743i)7-s + 0.333·9-s + (0.262 − 0.262i)11-s + 1.63i·13-s + (−0.0805 + 0.571i)15-s + (0.653 + 0.653i)17-s + (−0.550 + 0.550i)19-s + (0.429 + 0.429i)21-s + (−0.550 + 0.550i)23-s + (−0.961 − 0.276i)25-s + 0.192·27-s + (−1.38 − 1.38i)29-s − 1.38i·31-s + ⋯ |
Λ(s)=(=(1920s/2ΓC(s)L(s)(−0.301−0.953i)Λ(2−s)
Λ(s)=(=(1920s/2ΓC(s+1/2)L(s)(−0.301−0.953i)Λ(1−s)
Degree: |
2 |
Conductor: |
1920
= 27⋅3⋅5
|
Sign: |
−0.301−0.953i
|
Analytic conductor: |
15.3312 |
Root analytic conductor: |
3.91551 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1920(1567,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1920, ( :1/2), −0.301−0.953i)
|
Particular Values
L(1) |
≈ |
2.082583996 |
L(21) |
≈ |
2.082583996 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−T |
| 5 | 1+(0.311−2.21i)T |
good | 7 | 1+(−1.96−1.96i)T+7iT2 |
| 11 | 1+(−0.870+0.870i)T−11iT2 |
| 13 | 1−5.88iT−13T2 |
| 17 | 1+(−2.69−2.69i)T+17iT2 |
| 19 | 1+(2.40−2.40i)T−19iT2 |
| 23 | 1+(2.63−2.63i)T−23iT2 |
| 29 | 1+(7.43+7.43i)T+29iT2 |
| 31 | 1+7.72iT−31T2 |
| 37 | 1−4.49iT−37T2 |
| 41 | 1+4.84iT−41T2 |
| 43 | 1−0.461iT−43T2 |
| 47 | 1+(−4.66+4.66i)T−47iT2 |
| 53 | 1−2.41T+53T2 |
| 59 | 1+(−6.47−6.47i)T+59iT2 |
| 61 | 1+(8.50−8.50i)T−61iT2 |
| 67 | 1−6.40iT−67T2 |
| 71 | 1−13.3T+71T2 |
| 73 | 1+(1.62+1.62i)T+73iT2 |
| 79 | 1−4.14T+79T2 |
| 83 | 1+0.241T+83T2 |
| 89 | 1−2.86T+89T2 |
| 97 | 1+(−3.18−3.18i)T+97iT2 |
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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
−9.420032613571115075297725589612, −8.593748266414481993129357610866, −7.899969414326850085837943906881, −7.22056773614331699060815225508, −6.22328879621593566472550349639, −5.64238559063406775515756560508, −4.14237384624075078798192280513, −3.77945467851532103676629720558, −2.34800771790981318951507319961, −1.84052127433078041385779762723,
0.70055403012347426823643630679, 1.73672441708686397074711775198, 3.10544295981811937155537504124, 4.00920495518528164198760317117, 4.92126303355386558689257256388, 5.45439514012007955592058482620, 6.80309985910141405161512634116, 7.74402341644673368911850434108, 8.031039859284266391062541763642, 8.942204768457092201738379497636