L(s) = 1 | + 10·2-s + 50·4-s − 50·5-s + 80·7-s + 160·8-s − 500·10-s + 200·11-s + 410·13-s + 800·14-s + 444·16-s − 470·17-s − 2.50e3·20-s + 2.00e3·22-s − 680·23-s + 1.87e3·25-s + 4.10e3·26-s + 4.00e3·28-s + 856·31-s + 1.88e3·32-s − 4.70e3·34-s − 4.00e3·35-s − 1.51e3·37-s − 8.00e3·40-s − 1.90e3·41-s − 2.44e3·43-s + 1.00e4·44-s − 6.80e3·46-s + ⋯ |
L(s) = 1 | + 5/2·2-s + 25/8·4-s − 2·5-s + 1.63·7-s + 5/2·8-s − 5·10-s + 1.65·11-s + 2.42·13-s + 4.08·14-s + 1.73·16-s − 1.62·17-s − 6.25·20-s + 4.13·22-s − 1.28·23-s + 3·25-s + 6.06·26-s + 5.10·28-s + 0.890·31-s + 1.83·32-s − 4.06·34-s − 3.26·35-s − 1.10·37-s − 5·40-s − 1.13·41-s − 1.31·43-s + 5.16·44-s − 3.21·46-s + ⋯ |
Λ(s)=(=(2025s/2ΓC(s)2L(s)Λ(5−s)
Λ(s)=(=(2025s/2ΓC(s+2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2025
= 34⋅52
|
Sign: |
1
|
Analytic conductor: |
21.6378 |
Root analytic conductor: |
2.15676 |
Motivic weight: |
4 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 2025, ( :2,2), 1)
|
Particular Values
L(25) |
≈ |
6.653285197 |
L(21) |
≈ |
6.653285197 |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1 | (1+p2T)2 |
good | 2 | C22 | 1−5pT+25pT2−5p5T3+p8T4 |
| 7 | C22 | 1−80T+3200T2−80p4T3+p8T4 |
| 11 | C2 | (1−100T+p4T2)2 |
| 13 | C22 | 1−410T+84050T2−410p4T3+p8T4 |
| 17 | C22 | 1+470T+110450T2+470p4T3+p8T4 |
| 19 | C22 | 1−255458T2+p8T4 |
| 23 | C22 | 1+680T+231200T2+680p4T3+p8T4 |
| 29 | C22 | 1−1212062T2+p8T4 |
| 31 | C2 | (1−428T+p4T2)2 |
| 37 | C22 | 1+1510T+1140050T2+1510p4T3+p8T4 |
| 41 | C2 | (1+950T+p4T2)2 |
| 43 | C22 | 1+2440T+2976800T2+2440p4T3+p8T4 |
| 47 | C22 | 1−640T+204800T2−640p4T3+p8T4 |
| 53 | C22 | 1+1010T+510050T2+1010p4T3+p8T4 |
| 59 | C22 | 1+15455278T2+p8T4 |
| 61 | C2 | (1+3808T+p4T2)2 |
| 67 | C22 | 1−680T+231200T2−680p4T3+p8T4 |
| 71 | C2 | (1−3400T+p4T2)2 |
| 73 | C22 | 1−830T+344450T2−830p4T3+p8T4 |
| 79 | C22 | 1−32580338T2+p8T4 |
| 83 | C22 | 1−1360T+924800T2−1360p4T3+p8T4 |
| 89 | C22 | 1−120421982T2+p8T4 |
| 97 | C22 | 1−3230T+5216450T2−3230p4T3+p8T4 |
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
−15.50133925240445655956690304286, −14.55184907174678637870667255942, −14.29068522875328337188023796455, −13.53701589646932624912826224300, −13.43245594929028086832868376588, −12.17659912175821356236754410929, −12.13149936916721339299296009676, −11.34315914346371022995727218886, −11.33027201352804955914988420686, −10.60438310806087367317294542635, −8.617773990576936694291157365692, −8.612532198335473653513341754148, −7.75145767661683963319486832515, −6.60452755635537267947732298830, −6.26719112180111702936452251074, −4.93693590769966922850743753299, −4.48070873773314034949703832178, −3.81635200927165566925464618260, −3.56099129289252301819949620407, −1.46122874547668141519099399848,
1.46122874547668141519099399848, 3.56099129289252301819949620407, 3.81635200927165566925464618260, 4.48070873773314034949703832178, 4.93693590769966922850743753299, 6.26719112180111702936452251074, 6.60452755635537267947732298830, 7.75145767661683963319486832515, 8.612532198335473653513341754148, 8.617773990576936694291157365692, 10.60438310806087367317294542635, 11.33027201352804955914988420686, 11.34315914346371022995727218886, 12.13149936916721339299296009676, 12.17659912175821356236754410929, 13.43245594929028086832868376588, 13.53701589646932624912826224300, 14.29068522875328337188023796455, 14.55184907174678637870667255942, 15.50133925240445655956690304286