L(s) = 1 | + (−1.45 + 0.940i)3-s + (−0.196 + 1.11i)5-s + (2.99 − 2.51i)7-s + (1.23 − 2.73i)9-s + (0.324 + 1.84i)11-s + (0.688 − 0.250i)13-s + (−0.760 − 1.80i)15-s + (−0.944 + 1.63i)17-s + (1.37 + 2.37i)19-s + (−1.99 + 6.47i)21-s + (4.46 + 3.74i)23-s + (3.49 + 1.27i)25-s + (0.782 + 5.13i)27-s + (4.99 + 1.81i)29-s + (−1.02 − 0.861i)31-s + ⋯ |
L(s) = 1 | + (−0.839 + 0.542i)3-s + (−0.0877 + 0.497i)5-s + (1.13 − 0.949i)7-s + (0.410 − 0.911i)9-s + (0.0979 + 0.555i)11-s + (0.190 − 0.0694i)13-s + (−0.196 − 0.465i)15-s + (−0.229 + 0.396i)17-s + (0.314 + 0.544i)19-s + (−0.434 + 1.41i)21-s + (0.930 + 0.781i)23-s + (0.699 + 0.254i)25-s + (0.150 + 0.988i)27-s + (0.928 + 0.337i)29-s + (−0.184 − 0.154i)31-s + ⋯ |
Λ(s)=(=(432s/2ΓC(s)L(s)(0.743−0.669i)Λ(2−s)
Λ(s)=(=(432s/2ΓC(s+1/2)L(s)(0.743−0.669i)Λ(1−s)
Degree: |
2 |
Conductor: |
432
= 24⋅33
|
Sign: |
0.743−0.669i
|
Analytic conductor: |
3.44953 |
Root analytic conductor: |
1.85729 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ432(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 432, ( :1/2), 0.743−0.669i)
|
Particular Values
L(1) |
≈ |
1.11968+0.429776i |
L(21) |
≈ |
1.11968+0.429776i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.45−0.940i)T |
good | 5 | 1+(0.196−1.11i)T+(−4.69−1.71i)T2 |
| 7 | 1+(−2.99+2.51i)T+(1.21−6.89i)T2 |
| 11 | 1+(−0.324−1.84i)T+(−10.3+3.76i)T2 |
| 13 | 1+(−0.688+0.250i)T+(9.95−8.35i)T2 |
| 17 | 1+(0.944−1.63i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.37−2.37i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−4.46−3.74i)T+(3.99+22.6i)T2 |
| 29 | 1+(−4.99−1.81i)T+(22.2+18.6i)T2 |
| 31 | 1+(1.02+0.861i)T+(5.38+30.5i)T2 |
| 37 | 1+(1.69−2.94i)T+(−18.5−32.0i)T2 |
| 41 | 1+(1.68−0.614i)T+(31.4−26.3i)T2 |
| 43 | 1+(0.873+4.95i)T+(−40.4+14.7i)T2 |
| 47 | 1+(1.30−1.09i)T+(8.16−46.2i)T2 |
| 53 | 1−2.84T+53T2 |
| 59 | 1+(−1.95+11.0i)T+(−55.4−20.1i)T2 |
| 61 | 1+(−4.00+3.36i)T+(10.5−60.0i)T2 |
| 67 | 1+(1.77−0.646i)T+(51.3−43.0i)T2 |
| 71 | 1+(6.09−10.5i)T+(−35.5−61.4i)T2 |
| 73 | 1+(4.94+8.56i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−11.6−4.22i)T+(60.5+50.7i)T2 |
| 83 | 1+(10.9+3.99i)T+(63.5+53.3i)T2 |
| 89 | 1+(−2.86−4.96i)T+(−44.5+77.0i)T2 |
| 97 | 1+(0.0596+0.338i)T+(−91.1+33.1i)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
−11.10047041941858806271154347718, −10.56376032892829431162107675645, −9.763945750588420521850004565172, −8.519202639335075244826865578614, −7.36199570764663661004537464415, −6.68039515523202993660838598976, −5.33621891069289972892059022064, −4.53287962754311816452090860443, −3.48585001159102816200903703958, −1.35491781042669014145476312652,
1.06976758893814698218807919890, 2.55182322370590435894195993184, 4.63368356682946590020536070439, 5.21844234629027587889706846666, 6.24442319537071030236421314043, 7.30141640757458762335848476501, 8.432431483713410146021008407528, 8.919597912690041725009589347154, 10.43903444338181741013050161255, 11.30238050615807118963712614371