L(s) = 1 | + (2.34 + 1.58i)2-s + (2.99 + 7.41i)4-s + 6.32i·5-s + 10·7-s + (−4.69 + 22.1i)8-s + (−10.0 + 14.8i)10-s + 37.9i·11-s − 59.3i·13-s + (23.4 + 15.8i)14-s + (−46.0 + 44.4i)16-s + 75.0·17-s − 118. i·19-s + (−46.9 + 18.9i)20-s + (−60.0 + 88.9i)22-s − 150.·23-s + ⋯ |
L(s) = 1 | + (0.829 + 0.559i)2-s + (0.374 + 0.927i)4-s + 0.565i·5-s + 0.539·7-s + (−0.207 + 0.978i)8-s + (−0.316 + 0.469i)10-s + 1.04i·11-s − 1.26i·13-s + (0.447 + 0.301i)14-s + (−0.718 + 0.695i)16-s + 1.07·17-s − 1.43i·19-s + (−0.524 + 0.212i)20-s + (−0.581 + 0.862i)22-s − 1.36·23-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(0.207−0.978i)Λ(4−s)
Λ(s)=(=(72s/2ΓC(s+3/2)L(s)(0.207−0.978i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
0.207−0.978i
|
Analytic conductor: |
4.24813 |
Root analytic conductor: |
2.06110 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(37,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :3/2), 0.207−0.978i)
|
Particular Values
L(2) |
≈ |
1.80157+1.45983i |
L(21) |
≈ |
1.80157+1.45983i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−2.34−1.58i)T |
| 3 | 1 |
good | 5 | 1−6.32iT−125T2 |
| 7 | 1−10T+343T2 |
| 11 | 1−37.9iT−1.33e3T2 |
| 13 | 1+59.3iT−2.19e3T2 |
| 17 | 1−75.0T+4.91e3T2 |
| 19 | 1+118.iT−6.85e3T2 |
| 23 | 1+150.T+1.21e4T2 |
| 29 | 1+246.iT−2.43e4T2 |
| 31 | 1−62T+2.97e4T2 |
| 37 | 1+59.3iT−5.06e4T2 |
| 41 | 1−375.T+6.89e4T2 |
| 43 | 1−118.iT−7.95e4T2 |
| 47 | 1+450.T+1.03e5T2 |
| 53 | 1+132.iT−1.48e5T2 |
| 59 | 1−733.iT−2.05e5T2 |
| 61 | 1−533.iT−2.26e5T2 |
| 67 | 1−711.iT−3.00e5T2 |
| 71 | 1+3.57e5T2 |
| 73 | 1−30T+3.89e5T2 |
| 79 | 1−94T+4.93e5T2 |
| 83 | 1+670.iT−5.71e5T2 |
| 89 | 1+750.T+7.04e5T2 |
| 97 | 1−130T+9.12e5T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−14.59295082809585425797273081948, −13.39523049191980695836516765197, −12.36765913693917785528208200610, −11.30047651215847142037537010946, −9.989467472383640914860837199538, −8.130305362130291609866836020559, −7.24447542527269233845541087206, −5.81955109009358951116203357083, −4.47617310568157483875422497403, −2.73163268129322810587188564436,
1.50441641866594343879103180379, 3.63486623278384392430495117248, 5.05073538265799350646493468185, 6.28495307285863210110482881597, 8.133353239978715158563256564496, 9.543421036720079125957776495723, 10.83026446818280908255491794529, 11.87157234577676164089054842018, 12.66108947453960742686785229257, 14.20076786442430033763483518657