L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.896 − 3.34i)5-s + (0.5 + 0.866i)7-s + (0.707 + 0.707i)8-s − 3.46·10-s + 3.86·11-s + (5.59 + 1.5i)13-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (2.63 − 0.707i)17-s + (−6.09 − 1.63i)19-s + (0.896 + 3.34i)20-s + (−0.999 − 3.73i)22-s + (−1.93 − 1.93i)23-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.400 − 1.49i)5-s + (0.188 + 0.327i)7-s + (0.249 + 0.249i)8-s − 1.09·10-s + 1.16·11-s + (1.55 + 0.416i)13-s + (0.188 − 0.188i)14-s + (0.125 − 0.216i)16-s + (0.640 − 0.171i)17-s + (−1.39 − 0.374i)19-s + (0.200 + 0.748i)20-s + (−0.213 − 0.795i)22-s + (−0.402 − 0.402i)23-s + ⋯ |
Λ(s)=(=(666s/2ΓC(s)L(s)(−0.200+0.979i)Λ(2−s)
Λ(s)=(=(666s/2ΓC(s+1/2)L(s)(−0.200+0.979i)Λ(1−s)
Degree: |
2 |
Conductor: |
666
= 2⋅32⋅37
|
Sign: |
−0.200+0.979i
|
Analytic conductor: |
5.31803 |
Root analytic conductor: |
2.30608 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ666(125,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 666, ( :1/2), −0.200+0.979i)
|
Particular Values
L(1) |
≈ |
0.975006−1.19496i |
L(21) |
≈ |
0.975006−1.19496i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.258+0.965i)T |
| 3 | 1 |
| 37 | 1+(−6+i)T |
good | 5 | 1+(−0.896+3.34i)T+(−4.33−2.5i)T2 |
| 7 | 1+(−0.5−0.866i)T+(−3.5+6.06i)T2 |
| 11 | 1−3.86T+11T2 |
| 13 | 1+(−5.59−1.5i)T+(11.2+6.5i)T2 |
| 17 | 1+(−2.63+0.707i)T+(14.7−8.5i)T2 |
| 19 | 1+(6.09+1.63i)T+(16.4+9.5i)T2 |
| 23 | 1+(1.93+1.93i)T+23iT2 |
| 29 | 1+(−0.138+0.138i)T−29iT2 |
| 31 | 1+(3.63+3.63i)T+31iT2 |
| 41 | 1+(−1.03−1.79i)T+(−20.5+35.5i)T2 |
| 43 | 1+(7.56−7.56i)T−43iT2 |
| 47 | 1+8.76iT−47T2 |
| 53 | 1+(2.44+1.41i)T+(26.5+45.8i)T2 |
| 59 | 1+(−4.57+1.22i)T+(51.0−29.5i)T2 |
| 61 | 1+(0.633−2.36i)T+(−52.8−30.5i)T2 |
| 67 | 1+(2.59−1.5i)T+(33.5−58.0i)T2 |
| 71 | 1+(−2.44+1.41i)T+(35.5−61.4i)T2 |
| 73 | 1−15.3iT−73T2 |
| 79 | 1+(0.133+0.0358i)T+(68.4+39.5i)T2 |
| 83 | 1+(13.9+8.05i)T+(41.5+71.8i)T2 |
| 89 | 1+(−2.17−8.10i)T+(−77.0+44.5i)T2 |
| 97 | 1+(−11.0+11.0i)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
−10.16640178908842972476069818883, −9.267499692998938293984522458715, −8.720843840711833018588999467457, −8.191156442233113839044465788357, −6.52897432267769697211291300420, −5.66044447749669263183459575233, −4.50867400796500222907956540066, −3.79500040730989043570827445766, −1.99495815044823884045262345047, −1.03412299896558318611596580019,
1.59551931832647791367798104305, 3.32007312215929481981711908625, 4.13305128061001138629158340862, 5.85430337269808468075217034119, 6.30261929821711036783742254049, 7.09242621592012087340183070463, 8.062327536319307150692906964659, 8.938400613705980116293424199673, 9.971861232235404984582057468382, 10.70811600348675740690534102785