L(s) = 1 | − 5·5-s + 35·9-s − 22·11-s + 136·19-s − 100·25-s − 520·29-s + 350·31-s − 760·41-s − 175·45-s + 610·49-s + 110·55-s + 286·59-s + 1.35e3·61-s + 2.07e3·71-s − 436·79-s + 496·81-s − 2.55e3·89-s − 680·95-s − 770·99-s + 1.27e3·101-s + 284·109-s + 363·121-s + 1.12e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.29·9-s − 0.603·11-s + 1.64·19-s − 4/5·25-s − 3.32·29-s + 2.02·31-s − 2.89·41-s − 0.579·45-s + 1.77·49-s + 0.269·55-s + 0.631·59-s + 2.83·61-s + 3.46·71-s − 0.620·79-s + 0.680·81-s − 3.04·89-s − 0.734·95-s − 0.781·99-s + 1.25·101-s + 0.249·109-s + 3/11·121-s + 0.804·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
Λ(s)=(=(48400s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(48400s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
48400
= 24⋅52⋅112
|
Sign: |
1
|
Analytic conductor: |
168.491 |
Root analytic conductor: |
3.60283 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 48400, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
2.035427755 |
L(21) |
≈ |
2.035427755 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C2 | 1+pT+p3T2 |
| 11 | C1 | (1+pT)2 |
good | 3 | C22 | 1−35T2+p6T4 |
| 7 | C2 | (1−36T+p3T2)(1+36T+p3T2) |
| 13 | C22 | 1+470T2+p6T4 |
| 17 | C22 | 1−9142T2+p6T4 |
| 19 | C2 | (1−68T+p3T2)2 |
| 23 | C22 | 1−10483T2+p6T4 |
| 29 | C2 | (1+260T+p3T2)2 |
| 31 | C2 | (1−175T+p3T2)2 |
| 37 | C22 | 1−72407T2+p6T4 |
| 41 | C2 | (1+380T+p3T2)2 |
| 43 | C22 | 1−65914T2+p6T4 |
| 47 | C22 | 1−114546T2+p6T4 |
| 53 | C22 | 1−92250T2+p6T4 |
| 59 | C2 | (1−143T+p3T2)2 |
| 61 | C2 | (1−676T+p3T2)2 |
| 67 | C22 | 1−323347T2+p6T4 |
| 71 | C2 | (1−1035T+p3T2)2 |
| 73 | C22 | 1−668290T2+p6T4 |
| 79 | C2 | (1+218T+p3T2)2 |
| 83 | C22 | 1−568330T2+p6T4 |
| 89 | C2 | (1+1279T+p3T2)2 |
| 97 | C22 | 1−1230095T2+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
−11.95074739280655888122812547081, −11.58639949252771081875024192292, −11.30284408933867571540769753388, −10.56376511680353018762772406797, −9.897713089129453427350787820338, −9.859277426808712231543834332675, −9.347860888262493976750365016387, −8.376785139584930230905656127960, −8.184744302757415598630857206314, −7.43236208204780404458438339723, −7.12897220904777658051349220430, −6.71616314121717887024652794718, −5.58230306706859417250584546049, −5.42010855150368838168576024934, −4.66051454224359381589732379373, −3.71069324037631876847393722610, −3.66923080155452296311962486922, −2.43516102729658480829965200191, −1.63036085856316864720049048689, −0.61625561810630477085879647291,
0.61625561810630477085879647291, 1.63036085856316864720049048689, 2.43516102729658480829965200191, 3.66923080155452296311962486922, 3.71069324037631876847393722610, 4.66051454224359381589732379373, 5.42010855150368838168576024934, 5.58230306706859417250584546049, 6.71616314121717887024652794718, 7.12897220904777658051349220430, 7.43236208204780404458438339723, 8.184744302757415598630857206314, 8.376785139584930230905656127960, 9.347860888262493976750365016387, 9.859277426808712231543834332675, 9.897713089129453427350787820338, 10.56376511680353018762772406797, 11.30284408933867571540769753388, 11.58639949252771081875024192292, 11.95074739280655888122812547081