L(s) = 1 | + (1.89 + 1.09i)2-s + (−0.895 + 1.55i)3-s + (1.39 + 2.41i)4-s + 2.18i·5-s + (−3.39 + 1.96i)6-s + 1.73i·8-s + (−0.104 − 0.180i)9-s + (−2.39 + 4.14i)10-s + (−1.10 − 0.637i)11-s − 4.99·12-s + (−3.5 + 0.866i)13-s + (−3.39 − 1.96i)15-s + (0.895 − 1.55i)16-s + (1.5 + 2.59i)17-s − 0.456i·18-s + (5.68 − 3.28i)19-s + ⋯ |
L(s) = 1 | + (1.34 + 0.773i)2-s + (−0.517 + 0.895i)3-s + (0.697 + 1.20i)4-s + 0.978i·5-s + (−1.38 + 0.800i)6-s + 0.612i·8-s + (−0.0347 − 0.0602i)9-s + (−0.757 + 1.31i)10-s + (−0.332 − 0.192i)11-s − 1.44·12-s + (−0.970 + 0.240i)13-s + (−0.876 − 0.506i)15-s + (0.223 − 0.387i)16-s + (0.363 + 0.630i)17-s − 0.107i·18-s + (1.30 − 0.753i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.964−0.265i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.964−0.265i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.964−0.265i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(589,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.964−0.265i)
|
Particular Values
L(1) |
≈ |
0.321450+2.38260i |
L(21) |
≈ |
0.321450+2.38260i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(3.5−0.866i)T |
good | 2 | 1+(−1.89−1.09i)T+(1+1.73i)T2 |
| 3 | 1+(0.895−1.55i)T+(−1.5−2.59i)T2 |
| 5 | 1−2.18iT−5T2 |
| 11 | 1+(1.10+0.637i)T+(5.5+9.52i)T2 |
| 17 | 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−5.68+3.28i)T+(9.5−16.4i)T2 |
| 23 | 1+(3.79−6.56i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−1.10+1.91i)T+(−14.5−25.1i)T2 |
| 31 | 1+8.66iT−31T2 |
| 37 | 1+(−6−3.46i)T+(18.5+32.0i)T2 |
| 41 | 1+(2.20+1.27i)T+(20.5+35.5i)T2 |
| 43 | 1+(−2.18−3.78i)T+(−21.5+37.2i)T2 |
| 47 | 1−4.28iT−47T2 |
| 53 | 1+12.1T+53T2 |
| 59 | 1+(−7.66+4.42i)T+(29.5−51.0i)T2 |
| 61 | 1+(−6.37−11.0i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−9.87−5.70i)T+(33.5+58.0i)T2 |
| 71 | 1+(0.791−0.456i)T+(35.5−61.4i)T2 |
| 73 | 1−3.46iT−73T2 |
| 79 | 1+6T+79T2 |
| 83 | 1+3.55iT−83T2 |
| 89 | 1+(−2.52−1.45i)T+(44.5+77.0i)T2 |
| 97 | 1+(−13.1+7.61i)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
−11.32516336337575174425655259730, −9.967549535086978706515310635288, −9.729257188213040172826199697249, −7.79232937711489151074910999446, −7.28464483994551031939350551727, −6.16473037672556940125161873214, −5.47607828098225386784987627837, −4.64977096967864172658224527500, −3.74252111441710993804958462428, −2.72149990093153078315780972755,
0.946768860521896181542105793764, 2.21130596328021549322311126868, 3.48497073732612541650986951648, 4.82805757096900479260107752463, 5.24256810482218738934865840794, 6.27429343856061941558013860965, 7.34150167552812145586146284221, 8.289297613920826988663679709257, 9.595415179604265629080156151602, 10.43663376236652240744401618521