L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 1.25i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.627 + 1.08i)10-s + (4.35 − 2.51i)11-s + (−3.43 + 1.08i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.26 − 3.91i)17-s + (−3.05 − 1.76i)19-s + (−1.08 − 0.627i)20-s + (−2.51 + 4.35i)22-s + (−2.56 − 4.44i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.561i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.198 + 0.343i)10-s + (1.31 − 0.757i)11-s + (−0.953 + 0.299i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.548 − 0.949i)17-s + (−0.700 − 0.404i)19-s + (−0.243 − 0.140i)20-s + (−0.535 + 0.928i)22-s + (−0.535 − 0.927i)23-s + ⋯ |
Λ(s)=(=(1638s/2ΓC(s)L(s)(0.287+0.957i)Λ(2−s)
Λ(s)=(=(1638s/2ΓC(s+1/2)L(s)(0.287+0.957i)Λ(1−s)
Degree: |
2 |
Conductor: |
1638
= 2⋅32⋅7⋅13
|
Sign: |
0.287+0.957i
|
Analytic conductor: |
13.0794 |
Root analytic conductor: |
3.61655 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1638(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1638, ( :1/2), 0.287+0.957i)
|
Particular Values
L(1) |
≈ |
1.124423492 |
L(21) |
≈ |
1.124423492 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1 |
| 7 | 1+(−0.866−0.5i)T |
| 13 | 1+(3.43−1.08i)T |
good | 5 | 1+1.25iT−5T2 |
| 11 | 1+(−4.35+2.51i)T+(5.5−9.52i)T2 |
| 17 | 1+(−2.26+3.91i)T+(−8.5−14.7i)T2 |
| 19 | 1+(3.05+1.76i)T+(9.5+16.4i)T2 |
| 23 | 1+(2.56+4.44i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−0.765−1.32i)T+(−14.5+25.1i)T2 |
| 31 | 1+0.439iT−31T2 |
| 37 | 1+(8.90−5.13i)T+(18.5−32.0i)T2 |
| 41 | 1+(−8.65+4.99i)T+(20.5−35.5i)T2 |
| 43 | 1+(1.83−3.17i)T+(−21.5−37.2i)T2 |
| 47 | 1−3.60iT−47T2 |
| 53 | 1+8.82T+53T2 |
| 59 | 1+(−0.542−0.313i)T+(29.5+51.0i)T2 |
| 61 | 1+(−4.49+7.79i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−6.40+3.69i)T+(33.5−58.0i)T2 |
| 71 | 1+(5.89+3.40i)T+(35.5+61.4i)T2 |
| 73 | 1+11.5iT−73T2 |
| 79 | 1+6.74T+79T2 |
| 83 | 1+9.57iT−83T2 |
| 89 | 1+(−7.28+4.20i)T+(44.5−77.0i)T2 |
| 97 | 1+(2.43+1.40i)T+(48.5+84.0i)T2 |
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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
−9.042303568097802109442459921568, −8.624307915318552219475469459179, −7.71939991882906642577516743846, −6.82404768351502607700422487191, −6.17809075574567712304197797033, −5.07913960842831024193309362748, −4.43998190289900610182154746102, −3.05834199374618760865495683949, −1.78751009987588553996505260975, −0.56020374019418857421822118974,
1.37566802431133529661062055954, 2.31384888307251556914184640873, 3.57724943123462653509897394998, 4.28305775953619292799277699795, 5.53979410659511598123027094602, 6.59678534884718205820060887593, 7.21642504759040086178011827870, 7.977081832223158844724820017107, 8.815250495253581740176245987184, 9.699613058796789436255358358449