L(s) = 1 | − 16·4-s − 1.19e3·13-s + 256·16-s + 2.20e3·17-s − 2.10e3·23-s + 3.64e3·25-s + 8.20e3·29-s + 1.99e4·43-s + 2.25e4·49-s + 1.91e4·52-s + 1.50e3·53-s − 1.15e5·61-s − 4.09e3·64-s − 3.52e4·68-s + 1.26e5·79-s + 3.36e4·92-s − 5.83e4·100-s + 2.26e5·101-s + 5.00e4·103-s − 4.98e4·107-s − 2.00e5·113-s − 1.31e5·116-s + 3.07e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.96·13-s + 1/4·16-s + 1.84·17-s − 0.827·23-s + 1.16·25-s + 1.81·29-s + 1.64·43-s + 1.34·49-s + 0.981·52-s + 0.0733·53-s − 3.98·61-s − 1/8·64-s − 0.923·68-s + 2.27·79-s + 0.413·92-s − 0.583·100-s + 2.20·101-s + 0.465·103-s − 0.420·107-s − 1.47·113-s − 0.906·116-s + 1.91·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
Λ(s)=(=(54756s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(54756s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
54756
= 22⋅34⋅132
|
Sign: |
1
|
Analytic conductor: |
1408.48 |
Root analytic conductor: |
6.12615 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 54756, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
2.181678003 |
L(21) |
≈ |
2.181678003 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1+p4T2 |
| 3 | | 1 |
| 13 | C2 | 1+92pT+p5T2 |
good | 5 | C22 | 1−3649T2+p10T4 |
| 7 | C22 | 1−461p2T2+p10T4 |
| 11 | C22 | 1−307702T2+p10T4 |
| 17 | C2 | (1−1101T+p5T2)2 |
| 19 | C22 | 1−3583298T2+p10T4 |
| 23 | C2 | (1+1050T+p5T2)2 |
| 29 | C2 | (1−4104T+p5T2)2 |
| 31 | C22 | 1+35363074T2+p10T4 |
| 37 | C22 | 1−62841233T2+p10T4 |
| 41 | C22 | 1−141842002T2+p10T4 |
| 43 | C2 | (1−9995T+p5T2)2 |
| 47 | C22 | 1−450028765T2+p10T4 |
| 53 | C2 | (1−750T+p5T2)2 |
| 59 | C22 | 1+246071246T2+p10T4 |
| 61 | C2 | (1+57920T+p5T2)2 |
| 67 | C22 | 1−2179862870T2+p10T4 |
| 71 | C22 | 1+454456379T2+p10T4 |
| 73 | C22 | 1−680937230T2+p10T4 |
| 79 | C2 | (1−63202T+p5T2)2 |
| 83 | C22 | 1−4802491522T2+p10T4 |
| 89 | C22 | 1−189689614T2+p10T4 |
| 97 | C22 | 1+8570163790T2+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
−12.01564722989896368056223299065, −10.85941592912127044059202774695, −10.41389127526647543608719727596, −10.23679384610399907726125006126, −9.502633645548166488651127554080, −9.302176830956230348269183314698, −8.662558588919773944038673621331, −7.911105876696710420724617482656, −7.69034022700876328380984609525, −7.20402634972305475145049259457, −6.46717429904714964533298894295, −5.84144785520744976759359268610, −5.31584219160445818996791528294, −4.63755667528941151464541650240, −4.41321701233527808421709477405, −3.33124898745015316029784352709, −2.88290570232400841982222788082, −2.14130213880402589254586535111, −1.07623588320045590845175062375, −0.50840447317880579821769245286,
0.50840447317880579821769245286, 1.07623588320045590845175062375, 2.14130213880402589254586535111, 2.88290570232400841982222788082, 3.33124898745015316029784352709, 4.41321701233527808421709477405, 4.63755667528941151464541650240, 5.31584219160445818996791528294, 5.84144785520744976759359268610, 6.46717429904714964533298894295, 7.20402634972305475145049259457, 7.69034022700876328380984609525, 7.911105876696710420724617482656, 8.662558588919773944038673621331, 9.302176830956230348269183314698, 9.502633645548166488651127554080, 10.23679384610399907726125006126, 10.41389127526647543608719727596, 10.85941592912127044059202774695, 12.01564722989896368056223299065