L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 0.732i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.366 + 0.633i)10-s + (−4.5 + 2.59i)11-s + (0.866 − 3.5i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.13 − 1.96i)17-s + (−0.401 − 0.232i)19-s + (−0.633 − 0.366i)20-s + (2.59 − 4.5i)22-s + (3.73 + 6.46i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.327i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.115 + 0.200i)10-s + (−1.35 + 0.783i)11-s + (0.240 − 0.970i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.275 − 0.476i)17-s + (−0.0922 − 0.0532i)19-s + (−0.141 − 0.0818i)20-s + (0.553 − 0.959i)22-s + (0.778 + 1.34i)23-s + ⋯ |
Λ(s)=(=(1638s/2ΓC(s)L(s)(0.967+0.252i)Λ(2−s)
Λ(s)=(=(1638s/2ΓC(s+1/2)L(s)(0.967+0.252i)Λ(1−s)
Degree: |
2 |
Conductor: |
1638
= 2⋅32⋅7⋅13
|
Sign: |
0.967+0.252i
|
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), 0.967+0.252i)
|
Particular Values
L(1) |
≈ |
1.163211644 |
L(21) |
≈ |
1.163211644 |
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+(−0.866+3.5i)T |
good | 5 | 1+0.732iT−5T2 |
| 11 | 1+(4.5−2.59i)T+(5.5−9.52i)T2 |
| 17 | 1+(−1.13+1.96i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.401+0.232i)T+(9.5+16.4i)T2 |
| 23 | 1+(−3.73−6.46i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.76+3.06i)T+(−14.5+25.1i)T2 |
| 31 | 1+3.26iT−31T2 |
| 37 | 1+(−5.83+3.36i)T+(18.5−32.0i)T2 |
| 41 | 1+(−7.33+4.23i)T+(20.5−35.5i)T2 |
| 43 | 1+(4.36−7.56i)T+(−21.5−37.2i)T2 |
| 47 | 1+3.92iT−47T2 |
| 53 | 1−9.92T+53T2 |
| 59 | 1+(7.73+4.46i)T+(29.5+51.0i)T2 |
| 61 | 1+(1.86−3.23i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−8.66+5i)T+(33.5−58.0i)T2 |
| 71 | 1+(6.29+3.63i)T+(35.5+61.4i)T2 |
| 73 | 1+1.46iT−73T2 |
| 79 | 1−9T+79T2 |
| 83 | 1+9.26iT−83T2 |
| 89 | 1+(−14.5+8.42i)T+(44.5−77.0i)T2 |
| 97 | 1+(12.2+7.09i)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
−9.351550597979900169783779023046, −8.492683364541132851426200725972, −7.59847621732030398201762062895, −7.43334647281825896936716202728, −6.02106453341447072626562259873, −5.32613991654985365664497739295, −4.65619463482084380411229580780, −3.14427427841601569963995780953, −2.12903865573464311449534395692, −0.70681676093335654062076323899,
0.975709295342223765033876275038, 2.36396898067471465110391944411, 3.16404193596859141302802824535, 4.32196959623544973508580373371, 5.29234027864566441476587721927, 6.38418109883951412317221690378, 7.10550817602247635044449926623, 8.041971325360596332343202525212, 8.585382099308550828999362735459, 9.346250681074585387352784394507