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: |
2
|
Selberg data: |
(4, 5308416, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
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
−8.168167927255731955086518353937, −8.151983544079724472110008609149, −7.77092133738538295394211340733, −7.48925166552715238872542985600, −6.84013548926934587914514500904, −6.62444761569389701250182031264, −6.05368044855299303696892849446, −5.85081188139248230707909492016, −5.20150023306796514361114272350, −4.92522210563609866280458026059, −4.17708255379714852694468200092, −4.09953707129669894010031102254, −3.57012330100525943800995470349, −3.23067451718472970834016554478, −2.60282025855601100967287450490, −2.09107140383675182861094831733, −1.35637473763195318488072520121, −1.01339538556350284911064830202, 0, 0,
1.01339538556350284911064830202, 1.35637473763195318488072520121, 2.09107140383675182861094831733, 2.60282025855601100967287450490, 3.23067451718472970834016554478, 3.57012330100525943800995470349, 4.09953707129669894010031102254, 4.17708255379714852694468200092, 4.92522210563609866280458026059, 5.20150023306796514361114272350, 5.85081188139248230707909492016, 6.05368044855299303696892849446, 6.62444761569389701250182031264, 6.84013548926934587914514500904, 7.48925166552715238872542985600, 7.77092133738538295394211340733, 8.151983544079724472110008609149, 8.168167927255731955086518353937