L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + 0.732i·5-s + (0.866 − 0.499i)6-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.366 − 0.633i)10-s + (1.5 + 0.866i)11-s − 0.999·12-s + (1.59 − 3.23i)13-s − 0.999·14-s + (−0.633 − 0.366i)15-s + (−0.5 + 0.866i)16-s + (1.86 + 3.23i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 0.327i·5-s + (0.353 − 0.204i)6-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.115 − 0.200i)10-s + (0.452 + 0.261i)11-s − 0.288·12-s + (0.443 − 0.896i)13-s − 0.267·14-s + (−0.163 − 0.0945i)15-s + (−0.125 + 0.216i)16-s + (0.452 + 0.783i)17-s + ⋯ |
Λ(s)=(=(546s/2ΓC(s)L(s)(0.824−0.565i)Λ(2−s)
Λ(s)=(=(546s/2ΓC(s+1/2)L(s)(0.824−0.565i)Λ(1−s)
Degree: |
2 |
Conductor: |
546
= 2⋅3⋅7⋅13
|
Sign: |
0.824−0.565i
|
Analytic conductor: |
4.35983 |
Root analytic conductor: |
2.08802 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ546(43,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 546, ( :1/2), 0.824−0.565i)
|
Particular Values
L(1) |
≈ |
0.990612+0.306880i |
L(21) |
≈ |
0.990612+0.306880i |
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+(0.5−0.866i)T |
| 7 | 1+(−0.866+0.5i)T |
| 13 | 1+(−1.59+3.23i)T |
good | 5 | 1−0.732iT−5T2 |
| 11 | 1+(−1.5−0.866i)T+(5.5+9.52i)T2 |
| 17 | 1+(−1.86−3.23i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−0.866+0.5i)T+(9.5−16.4i)T2 |
| 23 | 1+(1.73−3i)T+(−11.5−19.9i)T2 |
| 29 | 1+(3.23−5.59i)T+(−14.5−25.1i)T2 |
| 31 | 1−2.19iT−31T2 |
| 37 | 1+(−5.83−3.36i)T+(18.5+32.0i)T2 |
| 41 | 1+(−2.59−1.5i)T+(20.5+35.5i)T2 |
| 43 | 1+(−1.63−2.83i)T+(−21.5+37.2i)T2 |
| 47 | 1−2.46iT−47T2 |
| 53 | 1−7T+53T2 |
| 59 | 1+(−0.803+0.464i)T+(29.5−51.0i)T2 |
| 61 | 1+(2.59+4.5i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−7.73−4.46i)T+(33.5+58.0i)T2 |
| 71 | 1+(1.90−1.09i)T+(35.5−61.4i)T2 |
| 73 | 1+5.46iT−73T2 |
| 79 | 1−2.07T+79T2 |
| 83 | 1−0.196iT−83T2 |
| 89 | 1+(9.06+5.23i)T+(44.5+77.0i)T2 |
| 97 | 1+(−13.5+7.83i)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
−10.79138033454868803260646687652, −10.14101182143248310448590899383, −9.265719529422004260797475720291, −8.328420958027490393525284725526, −7.45635512146812402965429340057, −6.37243157379974201728589590801, −5.30416635511018551316427233145, −4.00716912079894669915291008645, −3.02685845403989968944866945151, −1.29360346199130214710424497254,
0.926301273700834640319346484287, 2.32096330944653758915800400891, 4.14794305922382447816013905362, 5.38567503252472026963338892338, 6.25984439166830131523684117458, 7.15873600195455352732704127888, 8.049184739844742437751944333469, 8.910811064175830418972629874358, 9.607515564867724926863919977306, 10.79085251521530519987053866857