L(s) = 1 | + 6.25i·2-s − 21.3i·3-s − 7.11·4-s + 133.·6-s − 116. i·7-s + 155. i·8-s − 213.·9-s − 259.·11-s + 152. i·12-s + 169i·13-s + 727.·14-s − 1.20e3·16-s − 876. i·17-s − 1.33e3i·18-s + 921.·19-s + ⋯ |
L(s) = 1 | + 1.10i·2-s − 1.37i·3-s − 0.222·4-s + 1.51·6-s − 0.897i·7-s + 0.859i·8-s − 0.880·9-s − 0.645·11-s + 0.305i·12-s + 0.277i·13-s + 0.992·14-s − 1.17·16-s − 0.735i·17-s − 0.973i·18-s + 0.585·19-s + ⋯ |
Λ(s)=(=(325s/2ΓC(s)L(s)(−0.894+0.447i)Λ(6−s)
Λ(s)=(=(325s/2ΓC(s+5/2)L(s)(−0.894+0.447i)Λ(1−s)
Degree: |
2 |
Conductor: |
325
= 52⋅13
|
Sign: |
−0.894+0.447i
|
Analytic conductor: |
52.1247 |
Root analytic conductor: |
7.21974 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ325(274,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 325, ( :5/2), −0.894+0.447i)
|
Particular Values
L(3) |
≈ |
0.5452958553 |
L(21) |
≈ |
0.5452958553 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 13 | 1−169iT |
good | 2 | 1−6.25iT−32T2 |
| 3 | 1+21.3iT−243T2 |
| 7 | 1+116.iT−1.68e4T2 |
| 11 | 1+259.T+1.61e5T2 |
| 17 | 1+876.iT−1.41e6T2 |
| 19 | 1−921.T+2.47e6T2 |
| 23 | 1+2.05e3iT−6.43e6T2 |
| 29 | 1+781.T+2.05e7T2 |
| 31 | 1+5.80e3T+2.86e7T2 |
| 37 | 1−2.81e3iT−6.93e7T2 |
| 41 | 1+5.40e3T+1.15e8T2 |
| 43 | 1−5.18e3iT−1.47e8T2 |
| 47 | 1−9.66e3iT−2.29e8T2 |
| 53 | 1+763.iT−4.18e8T2 |
| 59 | 1+4.44e4T+7.14e8T2 |
| 61 | 1+5.08e4T+8.44e8T2 |
| 67 | 1−6.73e4iT−1.35e9T2 |
| 71 | 1+7.95e4T+1.80e9T2 |
| 73 | 1+5.51e4iT−2.07e9T2 |
| 79 | 1−6.64e4T+3.07e9T2 |
| 83 | 1−3.97e4iT−3.93e9T2 |
| 89 | 1+5.98e4T+5.58e9T2 |
| 97 | 1+4.16e4iT−8.58e9T2 |
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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
−10.46965591068427910740718575786, −9.045738676594913244141566878022, −7.87439955395656975289081220226, −7.42921783309942903585018455991, −6.75149923094796152809336867777, −5.83703545538146828027550499330, −4.65009716430432302613886380304, −2.78728132722754434436462039183, −1.49729564351410859940226877168, −0.13160402507789351749944687784,
1.74781311710972457969136815359, 2.99868734844195749244487282668, 3.76098703836354212517163431541, 4.98317167694896253399841079879, 5.91314089536925104845374446175, 7.52621151723656881452324087281, 8.919081058627703981566489061273, 9.536024327785707647445825287030, 10.43666230240860933508718463175, 10.92260517958753307560318100737