L(s) = 1 | + (−1.84 + 1.55i)2-s + (1.53 + 0.794i)3-s + (0.665 − 3.77i)4-s + (−4.07 + 0.920i)6-s + (0.0583 + 0.330i)7-s + (2.20 + 3.82i)8-s + (1.73 + 2.44i)9-s + (−4.97 − 1.81i)11-s + (4.01 − 5.27i)12-s + (4.68 + 3.93i)13-s + (−0.621 − 0.521i)14-s + (−2.82 − 1.02i)16-s + (−1.52 + 2.63i)17-s + (−7.01 − 1.82i)18-s + (−0.260 − 0.450i)19-s + ⋯ |
L(s) = 1 | + (−1.30 + 1.09i)2-s + (0.888 + 0.458i)3-s + (0.332 − 1.88i)4-s + (−1.66 + 0.375i)6-s + (0.0220 + 0.125i)7-s + (0.780 + 1.35i)8-s + (0.579 + 0.814i)9-s + (−1.49 − 0.545i)11-s + (1.16 − 1.52i)12-s + (1.29 + 1.09i)13-s + (−0.166 − 0.139i)14-s + (−0.706 − 0.257i)16-s + (−0.368 + 0.638i)17-s + (−1.65 − 0.429i)18-s + (−0.0597 − 0.103i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(−0.948−0.317i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(−0.948−0.317i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
−0.948−0.317i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(76,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), −0.948−0.317i)
|
Particular Values
L(1) |
≈ |
0.139164+0.853176i |
L(21) |
≈ |
0.139164+0.853176i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−1.53−0.794i)T |
| 5 | 1 |
good | 2 | 1+(1.84−1.55i)T+(0.347−1.96i)T2 |
| 7 | 1+(−0.0583−0.330i)T+(−6.57+2.39i)T2 |
| 11 | 1+(4.97+1.81i)T+(8.42+7.07i)T2 |
| 13 | 1+(−4.68−3.93i)T+(2.25+12.8i)T2 |
| 17 | 1+(1.52−2.63i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.260+0.450i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.0350−0.198i)T+(−21.6−7.86i)T2 |
| 29 | 1+(−1.41+1.18i)T+(5.03−28.5i)T2 |
| 31 | 1+(1.58−8.98i)T+(−29.1−10.6i)T2 |
| 37 | 1+(3.11−5.39i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−6.96−5.84i)T+(7.11+40.3i)T2 |
| 43 | 1+(6.38+2.32i)T+(32.9+27.6i)T2 |
| 47 | 1+(−2.11−12.0i)T+(−44.1+16.0i)T2 |
| 53 | 1−2.74T+53T2 |
| 59 | 1+(6.56−2.38i)T+(45.1−37.9i)T2 |
| 61 | 1+(2.16+12.2i)T+(−57.3+20.8i)T2 |
| 67 | 1+(4.29+3.60i)T+(11.6+65.9i)T2 |
| 71 | 1+(−1.58+2.74i)T+(−35.5−61.4i)T2 |
| 73 | 1+(−0.731−1.26i)T+(−36.5+63.2i)T2 |
| 79 | 1+(2.18−1.83i)T+(13.7−77.7i)T2 |
| 83 | 1+(0.578−0.485i)T+(14.4−81.7i)T2 |
| 89 | 1+(3.18+5.50i)T+(−44.5+77.0i)T2 |
| 97 | 1+(10.0+3.64i)T+(74.3+62.3i)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
−10.60933793944456939726355407132, −9.730141531995622623447402246353, −8.797164786093510767483140788615, −8.468905928219956551087251835405, −7.71635328858852406557908514966, −6.71713022162278874730062000481, −5.80629448787207396992071523114, −4.62722712410182522861614692333, −3.18117375301675183842217422090, −1.63609363472378331415879940455,
0.64409053044849407600159958944, 2.10149981282098793820612324571, 2.87422777887060052949276200218, 3.92325308700155062333517369886, 5.65241364020071873950786387431, 7.24030105396188949296801310490, 7.79927324365801666524053348024, 8.532514777441682370589947825041, 9.202720271134829870285345652158, 10.21984526915896140886591445704