L(s) = 1 | + (−1.93 − 1.11i)5-s + (−5.87 − 10.1i)7-s + (13.1 − 7.57i)11-s + (8.87 − 15.3i)13-s + 15.1i·17-s + 11.2·19-s + (29.2 + 16.8i)23-s + (2.5 + 4.33i)25-s + (−8.23 + 4.75i)29-s + (28.1 − 48.6i)31-s + 26.2i·35-s + 14·37-s + (22.5 + 12.9i)41-s + (−9.99 − 17.3i)43-s + (62.6 − 36.1i)47-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.223i)5-s + (−0.838 − 1.45i)7-s + (1.19 − 0.688i)11-s + (0.682 − 1.18i)13-s + 0.891i·17-s + 0.592·19-s + (1.27 + 0.733i)23-s + (0.100 + 0.173i)25-s + (−0.284 + 0.164i)29-s + (0.906 − 1.57i)31-s + 0.750i·35-s + 0.378·37-s + (0.548 + 0.316i)41-s + (−0.232 − 0.402i)43-s + (1.33 − 0.769i)47-s + ⋯ |
Λ(s)=(=(2160s/2ΓC(s)L(s)(−0.342+0.939i)Λ(3−s)
Λ(s)=(=(2160s/2ΓC(s+1)L(s)(−0.342+0.939i)Λ(1−s)
Degree: |
2 |
Conductor: |
2160
= 24⋅33⋅5
|
Sign: |
−0.342+0.939i
|
Analytic conductor: |
58.8557 |
Root analytic conductor: |
7.67174 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2160(881,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2160, ( :1), −0.342+0.939i)
|
Particular Values
L(23) |
≈ |
1.911784037 |
L(21) |
≈ |
1.911784037 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(1.93+1.11i)T |
good | 7 | 1+(5.87+10.1i)T+(−24.5+42.4i)T2 |
| 11 | 1+(−13.1+7.57i)T+(60.5−104.i)T2 |
| 13 | 1+(−8.87+15.3i)T+(−84.5−146.i)T2 |
| 17 | 1−15.1iT−289T2 |
| 19 | 1−11.2T+361T2 |
| 23 | 1+(−29.2−16.8i)T+(264.5+458.i)T2 |
| 29 | 1+(8.23−4.75i)T+(420.5−728.i)T2 |
| 31 | 1+(−28.1+48.6i)T+(−480.5−832.i)T2 |
| 37 | 1−14T+1.36e3T2 |
| 41 | 1+(−22.5−12.9i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(9.99+17.3i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+(−62.6+36.1i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1−37.6iT−2.80e3T2 |
| 59 | 1+(55.1+31.8i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(0.618+1.07i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(−42.9+74.4i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1−22.1iT−5.04e3T2 |
| 73 | 1+60.2T+5.32e3T2 |
| 79 | 1+(−51.6−89.4i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(78−45.0i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1+12.0iT−7.92e3T2 |
| 97 | 1+(49.8+86.3i)T+(−4.70e3+8.14e3i)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
−8.642891298667950451470598136360, −7.80905166839430325862377247309, −7.16979348368167677186954834400, −6.29994376629366258866840677016, −5.66108020847162495790946867432, −4.32392219935286735565107297109, −3.68675188433559511462929770764, −3.11702518747926252298455180037, −1.18156326639398668372176884846, −0.60777361732385959559167709955,
1.17707553305712283958066325555, 2.46351873647662955124014954851, 3.22910799283999924091780783616, 4.26676860046853419802175643567, 5.10938578662391252640132466635, 6.23815531205974755219894868323, 6.66583538966855005041215071839, 7.40579587854973827090713999654, 8.731117279391778808029654904573, 9.070987516444723734152399590674