L(s) = 1 | − 4·2-s + 7·3-s + 12·4-s − 28·6-s + 5·7-s − 32·8-s + 9·9-s − 51·11-s + 84·12-s − 26·13-s − 20·14-s + 80·16-s + 63·17-s − 36·18-s + 28·19-s + 35·21-s + 204·22-s + 9·23-s − 224·24-s + 104·26-s − 28·27-s + 60·28-s − 198·29-s + 175·31-s − 192·32-s − 357·33-s − 252·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.34·3-s + 3/2·4-s − 1.90·6-s + 0.269·7-s − 1.41·8-s + 1/3·9-s − 1.39·11-s + 2.02·12-s − 0.554·13-s − 0.381·14-s + 5/4·16-s + 0.898·17-s − 0.471·18-s + 0.338·19-s + 0.363·21-s + 1.97·22-s + 0.0815·23-s − 1.90·24-s + 0.784·26-s − 0.199·27-s + 0.404·28-s − 1.26·29-s + 1.01·31-s − 1.06·32-s − 1.88·33-s − 1.27·34-s + ⋯ |
Λ(s)=(=(422500s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(422500s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
422500
= 22⋅54⋅132
|
Sign: |
1
|
Analytic conductor: |
1470.81 |
Root analytic conductor: |
6.19283 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 422500, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
1.909535402 |
L(21) |
≈ |
1.909535402 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+pT)2 |
| 5 | | 1 |
| 13 | C1 | (1+pT)2 |
good | 3 | D4 | 1−7T+40T2−7p3T3+p6T4 |
| 7 | D4 | 1−5T+666T2−5p3T3+p6T4 |
| 11 | D4 | 1+51T+2656T2+51p3T3+p6T4 |
| 17 | D4 | 1−63T+9532T2−63p3T3+p6T4 |
| 19 | D4 | 1−28T+10134T2−28p3T3+p6T4 |
| 23 | D4 | 1−9T+16768T2−9p3T3+p6T4 |
| 29 | D4 | 1+198T+56899T2+198p3T3+p6T4 |
| 31 | D4 | 1−175T+64062T2−175p3T3+p6T4 |
| 37 | D4 | 1−632T+197382T2−632p3T3+p6T4 |
| 41 | D4 | 1−210T+136162T2−210p3T3+p6T4 |
| 43 | D4 | 1−23T+151560T2−23p3T3+p6T4 |
| 47 | D4 | 1−231T+129610T2−231p3T3+p6T4 |
| 53 | D4 | 1+186T+285823T2+186p3T3+p6T4 |
| 59 | D4 | 1−63T+410464T2−63p3T3+p6T4 |
| 61 | D4 | 1+182T+240063T2+182p3T3+p6T4 |
| 67 | D4 | 1−695T+523596T2−695p3T3+p6T4 |
| 71 | D4 | 1−390T−193778T2−390p3T3+p6T4 |
| 73 | D4 | 1−1526T+1231578T2−1526p3T3+p6T4 |
| 79 | D4 | 1+863T+991434T2+863p3T3+p6T4 |
| 83 | D4 | 1−315T+585604T2−315p3T3+p6T4 |
| 89 | D4 | 1+1638T+1580794T2+1638p3T3+p6T4 |
| 97 | D4 | 1−98T+996042T2−98p3T3+p6T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.14457539118605069147347330741, −9.668300201491275622154802080968, −9.426879682177825617975149046077, −9.323951962850405555687678826758, −8.344792590739361304099178245460, −8.249028020448431955275621115210, −7.88291219545671808973735537601, −7.72687333271058219638906277977, −7.11070658147207233406910946179, −6.63577798542684304804278507107, −5.72149247477053852037659481783, −5.71167017401484748595428607984, −4.82751247384841894313216212725, −4.27671325650200537056196266278, −3.17863618884491847534005883639, −3.17215838905741677826473202020, −2.29552048444035712697478301742, −2.23989294362442183115129447094, −1.12651675474771976048296918426, −0.49776152233045255375828449596,
0.49776152233045255375828449596, 1.12651675474771976048296918426, 2.23989294362442183115129447094, 2.29552048444035712697478301742, 3.17215838905741677826473202020, 3.17863618884491847534005883639, 4.27671325650200537056196266278, 4.82751247384841894313216212725, 5.71167017401484748595428607984, 5.72149247477053852037659481783, 6.63577798542684304804278507107, 7.11070658147207233406910946179, 7.72687333271058219638906277977, 7.88291219545671808973735537601, 8.249028020448431955275621115210, 8.344792590739361304099178245460, 9.323951962850405555687678826758, 9.426879682177825617975149046077, 9.668300201491275622154802080968, 10.14457539118605069147347330741