L(s) = 1 | + 19·3-s + 131·7-s + 225·9-s − 242·11-s + 1.61e3·13-s + 321·17-s + 3.43e3·19-s + 2.48e3·21-s − 4.53e3·23-s + 6.30e3·27-s − 5.03e3·29-s − 4.46e3·31-s − 4.59e3·33-s + 2.42e3·37-s + 3.05e4·39-s + 1.96e4·41-s + 1.49e4·43-s + 2.30e4·47-s − 9.73e3·49-s + 6.09e3·51-s + 7.20e3·53-s + 6.53e4·57-s + 5.08e4·59-s − 4.41e4·61-s + 2.94e4·63-s + 1.06e4·67-s − 8.61e4·69-s + ⋯ |
L(s) = 1 | + 1.21·3-s + 1.01·7-s + 0.925·9-s − 0.603·11-s + 2.64·13-s + 0.269·17-s + 2.18·19-s + 1.23·21-s − 1.78·23-s + 1.66·27-s − 1.11·29-s − 0.834·31-s − 0.734·33-s + 0.290·37-s + 3.22·39-s + 1.82·41-s + 1.22·43-s + 1.52·47-s − 0.579·49-s + 0.328·51-s + 0.352·53-s + 2.66·57-s + 1.90·59-s − 1.52·61-s + 0.935·63-s + 0.288·67-s − 2.17·69-s + ⋯ |
Λ(s)=(=(1210000s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(1210000s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1210000
= 24⋅54⋅112
|
Sign: |
1
|
Analytic conductor: |
31124.7 |
Root analytic conductor: |
13.2824 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1210000, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
11.60120995 |
L(21) |
≈ |
11.60120995 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | | 1 |
| 11 | C1 | (1+p2T)2 |
good | 3 | D4 | 1−19T+136T2−19p5T3+p10T4 |
| 7 | D4 | 1−131T+26898T2−131p5T3+p10T4 |
| 13 | D4 | 1−1610T+1247970T2−1610p5T3+p10T4 |
| 17 | D4 | 1−321T−1276838T2−321p5T3+p10T4 |
| 19 | D4 | 1−181pT+7887306T2−181p6T3+p10T4 |
| 23 | D4 | 1+4536T+14290234T2+4536p5T3+p10T4 |
| 29 | D4 | 1+5031T+38843968T2+5031p5T3+p10T4 |
| 31 | D4 | 1+4463T+56616342T2+4463p5T3+p10T4 |
| 37 | D4 | 1−2423T+88962936T2−2423p5T3+p10T4 |
| 41 | D4 | 1−19668T+301342822T2−19668p5T3+p10T4 |
| 43 | D4 | 1−14900T+309896886T2−14900p5T3+p10T4 |
| 47 | D4 | 1−23052T+368356594T2−23052p5T3+p10T4 |
| 53 | D4 | 1−7203T+763741528T2−7203p5T3+p10T4 |
| 59 | D4 | 1−50838T+1936923838T2−50838p5T3+p10T4 |
| 61 | D4 | 1+44177T+2028483204T2+44177p5T3+p10T4 |
| 67 | D4 | 1−10610T+2706695958T2−10610p5T3+p10T4 |
| 71 | D4 | 1+1089T+3235317082T2+1089p5T3+p10T4 |
| 73 | D4 | 1−92654T+6263930946T2−92654p5T3+p10T4 |
| 79 | D4 | 1+16334T+2724732846T2+16334p5T3+p10T4 |
| 83 | D4 | 1+88410T+7040835670T2+88410p5T3+p10T4 |
| 89 | D4 | 1+80817T+12691343080T2+80817p5T3+p10T4 |
| 97 | D4 | 1+102694T+19805064882T2+102694p5T3+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
−9.176894516032484860057997120476, −8.961364980937402005698639265930, −8.307029742220943211507899661529, −8.246946280354957252768157950035, −7.74778122326851753336330315412, −7.51677092283527981618101341444, −7.05942510600590970271029467172, −6.35670316316162626258771306644, −5.74128019242736873154683043532, −5.67445257104952971417818816104, −5.15918070736079742245679086494, −4.30307924368377175311274165247, −4.00770618111618236821536412602, −3.67458486274316711477940281776, −3.12734199964040000042329569679, −2.62303901742709195399658719509, −2.05775317636561767119593363302, −1.47677756649960857181038236173, −1.08351648550918211608498097422, −0.63125785332366393311861942954,
0.63125785332366393311861942954, 1.08351648550918211608498097422, 1.47677756649960857181038236173, 2.05775317636561767119593363302, 2.62303901742709195399658719509, 3.12734199964040000042329569679, 3.67458486274316711477940281776, 4.00770618111618236821536412602, 4.30307924368377175311274165247, 5.15918070736079742245679086494, 5.67445257104952971417818816104, 5.74128019242736873154683043532, 6.35670316316162626258771306644, 7.05942510600590970271029467172, 7.51677092283527981618101341444, 7.74778122326851753336330315412, 8.246946280354957252768157950035, 8.307029742220943211507899661529, 8.961364980937402005698639265930, 9.176894516032484860057997120476