L(s) = 1 | + (0.619 + 2.31i)2-s + (−1.28 − 1.16i)3-s + (−3.23 + 1.86i)4-s + (1.69 − 1.69i)5-s + (1.89 − 3.68i)6-s + (−1.36 − 0.366i)7-s + (−2.93 − 2.93i)8-s + (0.292 + 2.98i)9-s + (4.96 + 2.86i)10-s + (−1.69 + 0.453i)11-s + (6.31 + 1.36i)12-s + (−1.59 − 3.23i)13-s − 3.38i·14-s + (−4.14 + 0.202i)15-s + (1.23 − 2.13i)16-s + (1.07 + 1.85i)17-s + ⋯ |
L(s) = 1 | + (0.438 + 1.63i)2-s + (−0.740 − 0.671i)3-s + (−1.61 + 0.933i)4-s + (0.757 − 0.757i)5-s + (0.773 − 1.50i)6-s + (−0.516 − 0.138i)7-s + (−1.03 − 1.03i)8-s + (0.0975 + 0.995i)9-s + (1.56 + 0.906i)10-s + (−0.510 + 0.136i)11-s + (1.82 + 0.394i)12-s + (−0.443 − 0.896i)13-s − 0.904i·14-s + (−1.06 + 0.0523i)15-s + (0.308 − 0.533i)16-s + (0.260 + 0.450i)17-s + ⋯ |
Λ(s)=(=(39s/2ΓC(s)L(s)(0.248−0.968i)Λ(2−s)
Λ(s)=(=(39s/2ΓC(s+1/2)L(s)(0.248−0.968i)Λ(1−s)
Degree: |
2 |
Conductor: |
39
= 3⋅13
|
Sign: |
0.248−0.968i
|
Analytic conductor: |
0.311416 |
Root analytic conductor: |
0.558047 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ39(20,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 39, ( :1/2), 0.248−0.968i)
|
Particular Values
L(1) |
≈ |
0.603367+0.467915i |
L(21) |
≈ |
0.603367+0.467915i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.28+1.16i)T |
| 13 | 1+(1.59+3.23i)T |
good | 2 | 1+(−0.619−2.31i)T+(−1.73+i)T2 |
| 5 | 1+(−1.69+1.69i)T−5iT2 |
| 7 | 1+(1.36+0.366i)T+(6.06+3.5i)T2 |
| 11 | 1+(1.69−0.453i)T+(9.52−5.5i)T2 |
| 17 | 1+(−1.07−1.85i)T+(−8.5+14.7i)T2 |
| 19 | 1+(0.267−i)T+(−16.4−9.5i)T2 |
| 23 | 1+(−11.5−19.9i)T2 |
| 29 | 1+(−4.79−2.76i)T+(14.5+25.1i)T2 |
| 31 | 1+(−4.46−4.46i)T+31iT2 |
| 37 | 1+(1.76+6.59i)T+(−32.0+18.5i)T2 |
| 41 | 1+(0.166+0.619i)T+(−35.5+20.5i)T2 |
| 43 | 1+(−7.09+4.09i)T+(21.5−37.2i)T2 |
| 47 | 1+(6.77+6.77i)T+47iT2 |
| 53 | 1−4.62iT−53T2 |
| 59 | 1+(−1.23+4.62i)T+(−51.0−29.5i)T2 |
| 61 | 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 |
| 67 | 1+(8.46−2.26i)T+(58.0−33.5i)T2 |
| 71 | 1+(−4.62−1.23i)T+(61.4+35.5i)T2 |
| 73 | 1+(6.09−6.09i)T−73iT2 |
| 79 | 1−2T+79T2 |
| 83 | 1+(1.23−1.23i)T−83iT2 |
| 89 | 1+(9.70−2.60i)T+(77.0−44.5i)T2 |
| 97 | 1+(−3.36+12.5i)T+(−84.0−48.5i)T2 |
show more | |
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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
−16.54653774194307208799596870522, −15.66317440429096813659552138770, −14.16084729970142692144691609618, −13.10007655399432880274294286126, −12.52839326432450002874148299896, −10.24220840033534400824433356272, −8.461159458642994795156395746183, −7.19487632050671802615762527029, −5.91572767614074564480228031070, −5.04852373394208917471091899464,
2.85080616848113840615003351245, 4.66855553541343769555966900722, 6.32027748736098314606872610873, 9.527678998635192552840990833118, 10.12230220728853181358322335186, 11.20488791894176410864793618509, 12.15203152923105958253026705160, 13.44072305692369136671791587121, 14.51173279056174674504263381431, 16.04718524146681882632116795961