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) |
≈ |
0.1871032804 |
L(21) |
≈ |
0.1871032804 |
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
−8.674928589465623250389111698643, −8.368306148613313031431556298209, −8.074641098795012974045236367918, −7.77884339656101187089399900210, −7.29660493386083887982561372464, −7.06506671910476541982437116512, −6.40149049520788482817000551804, −6.37866287016735266993870994060, −5.57463690623759405273983903736, −5.34760665527239682916050832435, −4.64667176483337306434323193170, −4.57937812640227584449594731361, −3.88336965730274095748971935383, −3.73470580477882442327633158365, −2.87690499386731537393282823031, −2.86741936916663348175166069266, −1.94518721903970911902948190406, −1.63370186619493882920138533737, −0.795229420565376285503701111966, −0.10829014482775284369731266992,
0.10829014482775284369731266992, 0.795229420565376285503701111966, 1.63370186619493882920138533737, 1.94518721903970911902948190406, 2.86741936916663348175166069266, 2.87690499386731537393282823031, 3.73470580477882442327633158365, 3.88336965730274095748971935383, 4.57937812640227584449594731361, 4.64667176483337306434323193170, 5.34760665527239682916050832435, 5.57463690623759405273983903736, 6.37866287016735266993870994060, 6.40149049520788482817000551804, 7.06506671910476541982437116512, 7.29660493386083887982561372464, 7.77884339656101187089399900210, 8.074641098795012974045236367918, 8.368306148613313031431556298209, 8.674928589465623250389111698643