L(s) = 1 | − 4-s − 4·7-s + 6·13-s − 3·16-s + 6·19-s + 25-s + 4·28-s + 2·31-s − 8·37-s − 10·43-s + 2·49-s − 6·52-s + 4·61-s + 7·64-s − 14·67-s − 6·76-s + 4·79-s − 24·91-s + 2·97-s − 100-s + 28·103-s − 24·109-s + 12·112-s − 4·121-s − 2·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.51·7-s + 1.66·13-s − 3/4·16-s + 1.37·19-s + 1/5·25-s + 0.755·28-s + 0.359·31-s − 1.31·37-s − 1.52·43-s + 2/7·49-s − 0.832·52-s + 0.512·61-s + 7/8·64-s − 1.71·67-s − 0.688·76-s + 0.450·79-s − 2.51·91-s + 0.203·97-s − 0.0999·100-s + 2.75·103-s − 2.29·109-s + 1.13·112-s − 0.363·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Λ(s)=(=(731025s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(731025s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
731025
= 34⋅52⋅192
|
Sign: |
−1
|
Analytic conductor: |
46.6107 |
Root analytic conductor: |
2.61289 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 731025, ( :1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1×C1 | (1−T)(1+T) |
| 19 | C2 | 1−6T+pT2 |
good | 2 | C22 | 1+T2+p2T4 |
| 7 | C2×C2 | (1+pT2)(1+4T+pT2) |
| 11 | C22 | 1+4T2+p2T4 |
| 13 | C2×C2 | (1−4T+pT2)(1−2T+pT2) |
| 17 | C22 | 1+6T2+p2T4 |
| 23 | C22 | 1+40T2+p2T4 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2×C2 | (1−2T+pT2)(1+pT2) |
| 37 | C2×C2 | (1+2T+pT2)(1+6T+pT2) |
| 41 | C22 | 1−70T2+p2T4 |
| 43 | C2×C2 | (1+2T+pT2)(1+8T+pT2) |
| 47 | C22 | 1−32T2+p2T4 |
| 53 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 59 | C22 | 1−38T2+p2T4 |
| 61 | C2×C2 | (1−4T+pT2)(1+pT2) |
| 67 | C2×C2 | (1+4T+pT2)(1+10T+pT2) |
| 71 | C22 | 1−90T2+p2T4 |
| 73 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 79 | C2×C2 | (1−12T+pT2)(1+8T+pT2) |
| 83 | C22 | 1+44T2+p2T4 |
| 89 | C22 | 1−150T2+p2T4 |
| 97 | C2×C2 | (1−10T+pT2)(1+8T+pT2) |
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
−8.255360423462715209168018903284, −7.53475293787070780905295708732, −7.17181001068942514933126862250, −6.55837165840372060726346851759, −6.38338293152944269940879376425, −5.93171982463535917019017484152, −5.22033215255672620850904885032, −4.98598676245323315788210901850, −4.17106231067447187066318367227, −3.70382491597435883966281116650, −3.24923691385426228788415144395, −2.92157462896746393067989153495, −1.87831161997820156325930422244, −1.06581621034428724597756516792, 0,
1.06581621034428724597756516792, 1.87831161997820156325930422244, 2.92157462896746393067989153495, 3.24923691385426228788415144395, 3.70382491597435883966281116650, 4.17106231067447187066318367227, 4.98598676245323315788210901850, 5.22033215255672620850904885032, 5.93171982463535917019017484152, 6.38338293152944269940879376425, 6.55837165840372060726346851759, 7.17181001068942514933126862250, 7.53475293787070780905295708732, 8.255360423462715209168018903284