L(s) = 1 | + 18·3-s + 47·5-s + 243·9-s − 407·11-s + 449·13-s + 846·15-s − 1.86e3·17-s − 1.46e3·19-s − 44·23-s − 2.82e3·25-s + 2.91e3·27-s + 767·29-s − 1.11e4·31-s − 7.32e3·33-s − 3.11e3·37-s + 8.08e3·39-s − 7.84e3·41-s − 1.26e4·43-s + 1.14e4·45-s + 9.57e3·47-s − 3.36e4·51-s + 1.33e4·53-s − 1.91e4·55-s − 2.63e4·57-s − 4.75e4·59-s − 6.36e4·61-s + 2.11e4·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.840·5-s + 9-s − 1.01·11-s + 0.736·13-s + 0.970·15-s − 1.56·17-s − 0.929·19-s − 0.0173·23-s − 0.903·25-s + 0.769·27-s + 0.169·29-s − 2.08·31-s − 1.17·33-s − 0.373·37-s + 0.850·39-s − 0.728·41-s − 1.04·43-s + 0.840·45-s + 0.632·47-s − 1.81·51-s + 0.655·53-s − 0.852·55-s − 1.07·57-s − 1.77·59-s − 2.19·61-s + 0.619·65-s + ⋯ |
Λ(s)=(=(345744s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(345744s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
345744
= 24⋅32⋅74
|
Sign: |
1
|
Analytic conductor: |
8893.56 |
Root analytic conductor: |
9.71111 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 345744, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | (1−p2T)2 |
| 7 | | 1 |
good | 5 | D4 | 1−47T+5032T2−47p5T3+p10T4 |
| 11 | D4 | 1+37pT+361744T2+37p6T3+p10T4 |
| 13 | D4 | 1−449T+748730T2−449p5T3+p10T4 |
| 17 | D4 | 1+1868T+3683746T2+1868p5T3+p10T4 |
| 19 | D4 | 1+77pT+882870T2+77p6T3+p10T4 |
| 23 | D4 | 1+44T−4829330T2+44p5T3+p10T4 |
| 29 | D4 | 1−767T+20137030T2−767p5T3+p10T4 |
| 31 | D4 | 1+11170T+78225563T2+11170p5T3+p10T4 |
| 37 | D4 | 1+3113T+140329926T2+3113p5T3+p10T4 |
| 41 | D4 | 1+7842T+247022914T2+7842p5T3+p10T4 |
| 43 | D4 | 1+12629T+333109116T2+12629p5T3+p10T4 |
| 47 | D4 | 1−9576T+479320714T2−9576p5T3+p10T4 |
| 53 | D4 | 1−13395T+786785182T2−13395p5T3+p10T4 |
| 59 | D4 | 1+47521T+1225624018T2+47521p5T3+p10T4 |
| 61 | D4 | 1+63652T+2359366478T2+63652p5T3+p10T4 |
| 67 | D4 | 1−44541T+3121830628T2−44541p5T3+p10T4 |
| 71 | D4 | 1+125840T+7124822602T2+125840p5T3+p10T4 |
| 73 | D4 | 1−6039T+3156837796T2−6039p5T3+p10T4 |
| 79 | D4 | 1−17588T+1504957625T2−17588p5T3+p10T4 |
| 83 | D4 | 1−39325T+1663856032T2−39325p5T3+p10T4 |
| 89 | D4 | 1−83082T+12893200018T2−83082p5T3+p10T4 |
| 97 | D4 | 1−1905pT+241552312pT2−1905p6T3+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
−9.402108979184850373613484773864, −9.322367494442466004809934437308, −8.689590071441666485042366219094, −8.671521369871877751444029150487, −7.898167524291616087080901357767, −7.61964196192205606873634900900, −7.07432798677968991427662973035, −6.55529239654547634993881089896, −6.00914371027962963478502220060, −5.74868190524297496600881490007, −4.78133924744764873514058081633, −4.73019311604486794483657969696, −3.70060304537933683646842776416, −3.63558930581180187225155524084, −2.70351092046861496668028830351, −2.31172643502052337788849761468, −1.78400884788701691630491324446, −1.44267253025453914559143419610, 0, 0,
1.44267253025453914559143419610, 1.78400884788701691630491324446, 2.31172643502052337788849761468, 2.70351092046861496668028830351, 3.63558930581180187225155524084, 3.70060304537933683646842776416, 4.73019311604486794483657969696, 4.78133924744764873514058081633, 5.74868190524297496600881490007, 6.00914371027962963478502220060, 6.55529239654547634993881089896, 7.07432798677968991427662973035, 7.61964196192205606873634900900, 7.898167524291616087080901357767, 8.671521369871877751444029150487, 8.689590071441666485042366219094, 9.322367494442466004809934437308, 9.402108979184850373613484773864