L(s) = 1 | − 81·9-s + 432·11-s − 3.16e3·19-s + 9.13e3·29-s + 5.48e3·31-s − 2.67e4·41-s + 3.16e4·49-s + 3.11e4·59-s + 7.86e4·61-s + 1.14e5·71-s + 2.11e4·79-s + 6.56e3·81-s + 2.32e5·89-s − 3.49e4·99-s + 5.52e4·101-s + 4.60e5·109-s − 1.82e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.49e5·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 1.07·11-s − 2.00·19-s + 2.01·29-s + 1.02·31-s − 2.48·41-s + 1.88·49-s + 1.16·59-s + 2.70·61-s + 2.68·71-s + 0.380·79-s + 1/9·81-s + 3.11·89-s − 0.358·99-s + 0.538·101-s + 3.71·109-s − 1.13·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.403·169-s + ⋯ |
Λ(s)=(=(90000s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(90000s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
90000
= 24⋅32⋅54
|
Sign: |
1
|
Analytic conductor: |
2315.06 |
Root analytic conductor: |
6.93650 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 90000, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
3.129121456 |
L(21) |
≈ |
3.129121456 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C2 | 1+p4T2 |
| 5 | | 1 |
good | 7 | C22 | 1−31678T2+p10T4 |
| 11 | C2 | (1−216T+p5T2)2 |
| 13 | C22 | 1−149686T2+p10T4 |
| 17 | C22 | 1−2554558T2+p10T4 |
| 19 | C2 | (1+1580T+p5T2)2 |
| 23 | C22 | 1−4439470T2+p10T4 |
| 29 | C2 | (1−4566T+p5T2)2 |
| 31 | C2 | (1−2744T+p5T2)2 |
| 37 | C22 | 1−136608550T2+p10T4 |
| 41 | C2 | (1+13350T+p5T2)2 |
| 43 | C22 | 1+1960730T2+p10T4 |
| 47 | C22 | 1−341531038T2+p10T4 |
| 53 | C22 | 1−737547622T2+p10T4 |
| 59 | C2 | (1−264pT+p5T2)2 |
| 61 | C2 | (1−39302T+p5T2)2 |
| 67 | C22 | 1+412943402T2+p10T4 |
| 71 | C2 | (1−57120T+p5T2)2 |
| 73 | C22 | 1−1605781582T2+p10T4 |
| 79 | C2 | (1−10552T+p5T2)2 |
| 83 | C22 | 1+567290p2T2+p10T4 |
| 89 | C2 | (1−116430T+p5T2)2 |
| 97 | C22 | 1−17166940990T2+p10T4 |
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
−11.21115750836384368434329691411, −10.53104170324327817908202198584, −10.17441185710974251919933355535, −9.959430315134498845067613761964, −9.042912236779925476228196944217, −8.768100900965576323745367955383, −8.299413736454956642585167024553, −8.068771507561092723159468306081, −6.93576742330407891727936038341, −6.79751272198149287838526002146, −6.33720193390784163211742803727, −5.80699508266202579623123524944, −4.86707540992403090175228484694, −4.71558616395278519890037109159, −3.73206934808178501428819469534, −3.58330081840341827902651458639, −2.33615532441914227899475679760, −2.21704720789067257294825913843, −1.03548875318616564790633239603, −0.56064957615016467624518410478,
0.56064957615016467624518410478, 1.03548875318616564790633239603, 2.21704720789067257294825913843, 2.33615532441914227899475679760, 3.58330081840341827902651458639, 3.73206934808178501428819469534, 4.71558616395278519890037109159, 4.86707540992403090175228484694, 5.80699508266202579623123524944, 6.33720193390784163211742803727, 6.79751272198149287838526002146, 6.93576742330407891727936038341, 8.068771507561092723159468306081, 8.299413736454956642585167024553, 8.768100900965576323745367955383, 9.042912236779925476228196944217, 9.959430315134498845067613761964, 10.17441185710974251919933355535, 10.53104170324327817908202198584, 11.21115750836384368434329691411