L(s) = 1 | + 4-s − 4·7-s − 3·9-s − 3·16-s − 12·17-s − 4·28-s − 3·36-s + 4·37-s + 12·41-s + 16·43-s + 24·47-s + 9·49-s − 24·59-s + 12·63-s − 7·64-s − 16·67-s − 12·68-s + 16·79-s + 9·81-s + 12·89-s − 12·101-s − 4·109-s + 12·112-s + 48·119-s + 10·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1.51·7-s − 9-s − 3/4·16-s − 2.91·17-s − 0.755·28-s − 1/2·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s + 3.50·47-s + 9/7·49-s − 3.12·59-s + 1.51·63-s − 7/8·64-s − 1.95·67-s − 1.45·68-s + 1.80·79-s + 81-s + 1.27·89-s − 1.19·101-s − 0.383·109-s + 1.13·112-s + 4.40·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Λ(s)=(=(275625s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(275625s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
275625
= 32⋅54⋅72
|
Sign: |
1
|
Analytic conductor: |
17.5740 |
Root analytic conductor: |
2.04747 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 275625, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.8661560179 |
L(21) |
≈ |
0.8661560179 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | C2 | 1+pT2 |
| 5 | | 1 |
| 7 | C2 | 1+4T+pT2 |
good | 2 | C22 | 1−T2+p2T4 |
| 11 | C22 | 1−10T2+p2T4 |
| 13 | C2 | (1−pT2)2 |
| 17 | C2 | (1+6T+pT2)2 |
| 19 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 23 | C22 | 1−34T2+p2T4 |
| 29 | C22 | 1−10T2+p2T4 |
| 31 | C22 | 1−50T2+p2T4 |
| 37 | C2 | (1−2T+pT2)2 |
| 41 | C2 | (1−6T+pT2)2 |
| 43 | C2 | (1−8T+pT2)2 |
| 47 | C2 | (1−12T+pT2)2 |
| 53 | C2 | (1−pT2)2 |
| 59 | C2 | (1+12T+pT2)2 |
| 61 | C2 | (1−14T+pT2)(1+14T+pT2) |
| 67 | C2 | (1+8T+pT2)2 |
| 71 | C22 | 1−130T2+p2T4 |
| 73 | C22 | 1−98T2+p2T4 |
| 79 | C2 | (1−8T+pT2)2 |
| 83 | C2 | (1+pT2)2 |
| 89 | C2 | (1−6T+pT2)2 |
| 97 | C22 | 1−146T2+p2T4 |
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
−11.05728907308294597640692751467, −10.64505167473116839181775573867, −10.55601873144198600642825540133, −9.393858134678248784463624231204, −9.291608285219077657063013782677, −8.966822830681800343960945127234, −8.776884818319404234225133865568, −7.65404072485387587447613205576, −7.56293213253545126169148953859, −6.91856009216061670547587563037, −6.39309111447211168970554059232, −5.99747198042375773798681745579, −5.95496263296676107145589615451, −4.88773036380882085998920885317, −4.14485422861587002751360888280, −4.10825837911786170154677775303, −2.87131288274295921635037414432, −2.64352353327532264001546469447, −2.17070756404279439413224669342, −0.51127384343248679183538932567,
0.51127384343248679183538932567, 2.17070756404279439413224669342, 2.64352353327532264001546469447, 2.87131288274295921635037414432, 4.10825837911786170154677775303, 4.14485422861587002751360888280, 4.88773036380882085998920885317, 5.95496263296676107145589615451, 5.99747198042375773798681745579, 6.39309111447211168970554059232, 6.91856009216061670547587563037, 7.56293213253545126169148953859, 7.65404072485387587447613205576, 8.776884818319404234225133865568, 8.966822830681800343960945127234, 9.291608285219077657063013782677, 9.393858134678248784463624231204, 10.55601873144198600642825540133, 10.64505167473116839181775573867, 11.05728907308294597640692751467