L(s) = 1 | − 4·4-s + 5·9-s − 66·11-s + 16·16-s + 140·19-s + 450·29-s − 176·31-s − 20·36-s + 864·41-s + 264·44-s − 49·49-s − 960·59-s + 1.62e3·61-s − 64·64-s + 864·71-s − 560·76-s − 850·79-s − 704·81-s − 1.92e3·89-s − 330·99-s − 876·101-s + 3.83e3·109-s − 1.80e3·116-s + 605·121-s + 704·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 5/27·9-s − 1.80·11-s + 1/4·16-s + 1.69·19-s + 2.88·29-s − 1.01·31-s − 0.0925·36-s + 3.29·41-s + 0.904·44-s − 1/7·49-s − 2.11·59-s + 3.40·61-s − 1/8·64-s + 1.44·71-s − 0.845·76-s − 1.21·79-s − 0.965·81-s − 2.28·89-s − 0.335·99-s − 0.863·101-s + 3.36·109-s − 1.44·116-s + 5/11·121-s + 0.509·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
Λ(s)=(=(122500s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(122500s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
122500
= 22⋅54⋅72
|
Sign: |
1
|
Analytic conductor: |
426.450 |
Root analytic conductor: |
4.54430 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 122500, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
1.942514545 |
L(21) |
≈ |
1.942514545 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1+p2T2 |
| 5 | | 1 |
| 7 | C2 | 1+p2T2 |
good | 3 | C22 | 1−5T2+p6T4 |
| 11 | C2 | (1+3pT+p3T2)2 |
| 13 | C22 | 1−2545T2+p6T4 |
| 17 | C22 | 1+2495T2+p6T4 |
| 19 | C2 | (1−70T+p3T2)2 |
| 23 | C22 | 1−22570T2+p6T4 |
| 29 | C2 | (1−225T+p3T2)2 |
| 31 | C2 | (1+88T+p3T2)2 |
| 37 | C22 | 1−100150T2+p6T4 |
| 41 | C2 | (1−432T+p3T2)2 |
| 43 | C22 | 1−127330T2+p6T4 |
| 47 | C22 | 1−38725T2+p6T4 |
| 53 | C22 | 1+203510T2+p6T4 |
| 59 | C2 | (1+480T+p3T2)2 |
| 61 | C2 | (1−812T+p3T2)2 |
| 67 | C22 | 1−246310T2+p6T4 |
| 71 | C2 | (1−432T+p3T2)2 |
| 73 | C22 | 1−649870T2+p6T4 |
| 79 | C2 | (1+425T+p3T2)2 |
| 83 | C22 | 1−198790T2+p6T4 |
| 89 | C2 | (1+960T+p3T2)2 |
| 97 | C22 | 1−1322665T2+p6T4 |
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
−11.40985961231806188638372247154, −10.76966040685308886012835996517, −10.24476963401864994185354817956, −9.995752339047791671341062570221, −9.513617207004300963547223004915, −9.017475295201064056858708572042, −8.339658679850305248356127624637, −8.107674714999091627912678035565, −7.45313028963705608528013027553, −7.25563616634511887751888497145, −6.44844333451257791583636340482, −5.76265520629746189179065838504, −5.39350972483079320254787702001, −4.86675750534219107335004466489, −4.36993776404227476984938724357, −3.61767049061658995390571163022, −2.73033807496401543444987247950, −2.59403547171467179085322643698, −1.22113281003954753488476805137, −0.55752770705824908531302636196,
0.55752770705824908531302636196, 1.22113281003954753488476805137, 2.59403547171467179085322643698, 2.73033807496401543444987247950, 3.61767049061658995390571163022, 4.36993776404227476984938724357, 4.86675750534219107335004466489, 5.39350972483079320254787702001, 5.76265520629746189179065838504, 6.44844333451257791583636340482, 7.25563616634511887751888497145, 7.45313028963705608528013027553, 8.107674714999091627912678035565, 8.339658679850305248356127624637, 9.017475295201064056858708572042, 9.513617207004300963547223004915, 9.995752339047791671341062570221, 10.24476963401864994185354817956, 10.76966040685308886012835996517, 11.40985961231806188638372247154