L(s) = 1 | + 12·5-s − 40·13-s + 16·17-s − 142·25-s − 92·29-s + 328·37-s + 624·41-s − 238·49-s + 532·53-s + 264·61-s − 480·65-s + 492·73-s + 192·85-s + 2.78e3·89-s − 604·97-s + 2.93e3·101-s + 3.12e3·109-s − 3.10e3·113-s − 870·121-s − 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.07·5-s − 0.853·13-s + 0.228·17-s − 1.13·25-s − 0.589·29-s + 1.45·37-s + 2.37·41-s − 0.693·49-s + 1.37·53-s + 0.554·61-s − 0.915·65-s + 0.788·73-s + 0.245·85-s + 3.31·89-s − 0.632·97-s + 2.88·101-s + 2.74·109-s − 2.58·113-s − 0.653·121-s − 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
Λ(s)=(=(5308416s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(5308416s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
5308416
= 216⋅34
|
Sign: |
1
|
Analytic conductor: |
18479.7 |
Root analytic conductor: |
11.6593 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 5308416, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
5.051788571 |
L(21) |
≈ |
5.051788571 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
good | 5 | C2 | (1−6T+p3T2)2 |
| 7 | C22 | 1+34pT2+p6T4 |
| 11 | C22 | 1+870T2+p6T4 |
| 13 | C2 | (1+20T+p3T2)2 |
| 17 | C2 | (1−8T+p3T2)2 |
| 19 | C22 | 1+6550T2+p6T4 |
| 23 | C22 | 1−4338T2+p6T4 |
| 29 | C2 | (1+46T+p3T2)2 |
| 31 | C22 | 1+59134T2+p6T4 |
| 37 | C2 | (1−164T+p3T2)2 |
| 41 | C2 | (1−312T+p3T2)2 |
| 43 | C22 | 1−20186T2+p6T4 |
| 47 | C22 | 1+178974T2+p6T4 |
| 53 | C2 | (1−266T+p3T2)2 |
| 59 | C22 | 1+346246T2+p6T4 |
| 61 | C2 | (1−132T+p3T2)2 |
| 67 | C22 | 1+343478T2+p6T4 |
| 71 | C22 | 1+257070T2+p6T4 |
| 73 | C2 | (1−246T+p3T2)2 |
| 79 | C22 | 1+931870T2+p6T4 |
| 83 | C22 | 1+195606T2+p6T4 |
| 89 | C2 | (1−1392T+p3T2)2 |
| 97 | C2 | (1+302T+p3T2)2 |
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
−9.068320718430586707576346592331, −8.510203494958569075473569957636, −7.82914768500903149922454862135, −7.76071441139252348984212924569, −7.46117180224875122397741590603, −6.89088889203380375911102492517, −6.39493412954873162441148636136, −6.08390815882679726651876947523, −5.69092748653825105772375589094, −5.50236092771098475603889010913, −4.80189099312086770366939471157, −4.60339266337629697763503323342, −3.86413657686958596916412871263, −3.72186251088187267584349983865, −2.84899815356566449971456843329, −2.55299902155961116719969862621, −1.94524802376396156095771121162, −1.82057798586685079973966681706, −0.70976619479302057083419087927, −0.62323311141488764755729211682,
0.62323311141488764755729211682, 0.70976619479302057083419087927, 1.82057798586685079973966681706, 1.94524802376396156095771121162, 2.55299902155961116719969862621, 2.84899815356566449971456843329, 3.72186251088187267584349983865, 3.86413657686958596916412871263, 4.60339266337629697763503323342, 4.80189099312086770366939471157, 5.50236092771098475603889010913, 5.69092748653825105772375589094, 6.08390815882679726651876947523, 6.39493412954873162441148636136, 6.89088889203380375911102492517, 7.46117180224875122397741590603, 7.76071441139252348984212924569, 7.82914768500903149922454862135, 8.510203494958569075473569957636, 9.068320718430586707576346592331