L(s) = 1 | − 1.41i·2-s − 1.73·5-s + (1 − 2.44i)7-s − 2.82i·8-s + 2.44i·10-s − 1.41i·11-s − 2.44i·13-s + (−3.46 − 1.41i)14-s − 4.00·16-s + 5.19·17-s + 7.34i·19-s − 2.00·22-s + 2.82i·23-s − 2.00·25-s − 3.46·26-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 0.774·5-s + (0.377 − 0.925i)7-s − 0.999i·8-s + 0.774i·10-s − 0.426i·11-s − 0.679i·13-s + (−0.925 − 0.377i)14-s − 1.00·16-s + 1.26·17-s + 1.68i·19-s − 0.426·22-s + 0.589i·23-s − 0.400·25-s − 0.679·26-s + ⋯ |
Λ(s)=(=(189s/2ΓC(s)L(s)(−0.377+0.925i)Λ(2−s)
Λ(s)=(=(189s/2ΓC(s+1/2)L(s)(−0.377+0.925i)Λ(1−s)
Degree: |
2 |
Conductor: |
189
= 33⋅7
|
Sign: |
−0.377+0.925i
|
Analytic conductor: |
1.50917 |
Root analytic conductor: |
1.22848 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ189(188,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 189, ( :1/2), −0.377+0.925i)
|
Particular Values
L(1) |
≈ |
0.658358−0.979882i |
L(21) |
≈ |
0.658358−0.979882i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(−1+2.44i)T |
good | 2 | 1+1.41iT−2T2 |
| 5 | 1+1.73T+5T2 |
| 11 | 1+1.41iT−11T2 |
| 13 | 1+2.44iT−13T2 |
| 17 | 1−5.19T+17T2 |
| 19 | 1−7.34iT−19T2 |
| 23 | 1−2.82iT−23T2 |
| 29 | 1−7.07iT−29T2 |
| 31 | 1+2.44iT−31T2 |
| 37 | 1−5T+37T2 |
| 41 | 1−8.66T+41T2 |
| 43 | 1−5T+43T2 |
| 47 | 1+8.66T+47T2 |
| 53 | 1−11.3iT−53T2 |
| 59 | 1−8.66T+59T2 |
| 61 | 1−2.44iT−61T2 |
| 67 | 1−2T+67T2 |
| 71 | 1+14.1iT−71T2 |
| 73 | 1−73T2 |
| 79 | 1+13T+79T2 |
| 83 | 1−1.73T+83T2 |
| 89 | 1+10.3T+89T2 |
| 97 | 1−17.1iT−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
−12.13141041694563760704111852812, −11.25401025099405174551874117609, −10.52010450562826378801543801606, −9.700248769112340950507436858839, −8.006987996081415846114091344901, −7.42781759225687197034043682863, −5.81826367315714878428283556666, −4.06322499893154788370364151799, −3.26072537580287945895711806105, −1.21390625503354902557172281617,
2.50285706865570168890423261532, 4.47567587387250422591413427531, 5.60233471199409282489903453001, 6.75682140315435819717536517767, 7.72969249655452372306215068052, 8.507092058131594793711533661069, 9.639994425197906743321190836037, 11.33486793844748845137857339627, 11.67120188013919176218549325931, 12.83217219130144172886084074141