L(s) = 1 | + 2.10·3-s + (−2.18 − 0.478i)5-s + (1.94 − 1.94i)7-s + 1.41·9-s + (0.883 − 0.883i)11-s + (2.69 − 2.69i)13-s + (−4.58 − 1.00i)15-s − 0.326i·17-s + (−5.28 − 5.28i)19-s + (4.09 − 4.09i)21-s + (−0.557 − 0.557i)23-s + (4.54 + 2.09i)25-s − 3.33·27-s + (−1.35 + 5.21i)29-s + (3.27 − 3.27i)31-s + ⋯ |
L(s) = 1 | + 1.21·3-s + (−0.976 − 0.214i)5-s + (0.736 − 0.736i)7-s + 0.471·9-s + (0.266 − 0.266i)11-s + (0.747 − 0.747i)13-s + (−1.18 − 0.259i)15-s − 0.0792i·17-s + (−1.21 − 1.21i)19-s + (0.893 − 0.893i)21-s + (−0.116 − 0.116i)23-s + (0.908 + 0.418i)25-s − 0.641·27-s + (−0.252 + 0.967i)29-s + (0.588 − 0.588i)31-s + ⋯ |
Λ(s)=(=(1160s/2ΓC(s)L(s)(0.348+0.937i)Λ(2−s)
Λ(s)=(=(1160s/2ΓC(s+1/2)L(s)(0.348+0.937i)Λ(1−s)
Degree: |
2 |
Conductor: |
1160
= 23⋅5⋅29
|
Sign: |
0.348+0.937i
|
Analytic conductor: |
9.26264 |
Root analytic conductor: |
3.04345 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1160(713,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1160, ( :1/2), 0.348+0.937i)
|
Particular Values
L(1) |
≈ |
2.110666508 |
L(21) |
≈ |
2.110666508 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(2.18+0.478i)T |
| 29 | 1+(1.35−5.21i)T |
good | 3 | 1−2.10T+3T2 |
| 7 | 1+(−1.94+1.94i)T−7iT2 |
| 11 | 1+(−0.883+0.883i)T−11iT2 |
| 13 | 1+(−2.69+2.69i)T−13iT2 |
| 17 | 1+0.326iT−17T2 |
| 19 | 1+(5.28+5.28i)T+19iT2 |
| 23 | 1+(0.557+0.557i)T+23iT2 |
| 31 | 1+(−3.27+3.27i)T−31iT2 |
| 37 | 1−6.69T+37T2 |
| 41 | 1+(6.35+6.35i)T+41iT2 |
| 43 | 1+2.44T+43T2 |
| 47 | 1−11.9T+47T2 |
| 53 | 1+(−1.79−1.79i)T+53iT2 |
| 59 | 1+6.73iT−59T2 |
| 61 | 1+(−5.50+5.50i)T−61iT2 |
| 67 | 1+(−3.12−3.12i)T+67iT2 |
| 71 | 1+3.62iT−71T2 |
| 73 | 1+11.1iT−73T2 |
| 79 | 1+(−5.17−5.17i)T+79iT2 |
| 83 | 1+(−3.60−3.60i)T+83iT2 |
| 89 | 1+(−10.8−10.8i)T+89iT2 |
| 97 | 1+3.89T+97T2 |
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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
−9.299463410468743271056410213659, −8.631381767314844192674889231725, −8.109448817202197421017328702972, −7.48340017791655134561201526292, −6.52131678105717021591850271786, −5.10256117341001122114583804376, −4.11818273547936216856812287205, −3.51872243692626643299599230570, −2.38132045155952489346617767103, −0.826707141795315107263985317847,
1.71866978040072504220136216780, 2.68894184871111146730505059729, 3.85516274642118343481263558149, 4.35440752662396868624274367060, 5.79002492895016118402652923669, 6.77582961032013623431286606667, 7.84446523544885539107334547150, 8.374529713219187162936773118439, 8.771400886931038899490712356891, 9.774811896197410877634230820611