L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 8·11-s − 12-s − 4·13-s + 16-s − 18-s − 8·22-s + 24-s − 6·25-s + 4·26-s − 27-s − 32-s − 8·33-s + 36-s − 4·37-s + 4·39-s + 8·44-s − 48-s − 14·49-s + 6·50-s − 4·52-s + 54-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 1.70·22-s + 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s − 0.176·32-s − 1.39·33-s + 1/6·36-s − 0.657·37-s + 0.640·39-s + 1.20·44-s − 0.144·48-s − 2·49-s + 0.848·50-s − 0.554·52-s + 0.136·54-s + ⋯ |
Λ(s)=(=(249696s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(249696s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
249696
= 25⋅33⋅172
|
Sign: |
−1
|
Analytic conductor: |
15.9208 |
Root analytic conductor: |
1.99752 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 249696, ( :1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | 1+T |
| 3 | C1 | 1+T |
| 17 | C1×C1 | (1−T)(1+T) |
good | 5 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 7 | C2 | (1+pT2)2 |
| 11 | C2 | (1−4T+pT2)2 |
| 13 | C2 | (1+2T+pT2)2 |
| 19 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 31 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 37 | C2 | (1+2T+pT2)2 |
| 41 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 43 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 47 | C2 | (1+pT2)2 |
| 53 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 59 | C2 | (1+12T+pT2)2 |
| 61 | C2 | (1+10T+pT2)2 |
| 67 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1−10T+pT2)2 |
| 79 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 83 | C2 | (1+4T+pT2)2 |
| 89 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 97 | C2 | (1+14T+pT2)2 |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.080746654144001493159861980940, −8.000937863938982294341146593643, −7.972624440390951709752181868622, −7.22142790948264379459326388401, −6.79533660929922109224513153399, −6.37869009161194820234692371426, −6.05886413503827459123865903257, −5.36244298721167160718725340278, −4.66119827871442481374467622682, −4.24541437163957543556604196065, −3.56656081546400251035965478912, −2.92323665285260202872977027719, −1.78155952274398559703453953653, −1.43438626728056627432143840048, 0,
1.43438626728056627432143840048, 1.78155952274398559703453953653, 2.92323665285260202872977027719, 3.56656081546400251035965478912, 4.24541437163957543556604196065, 4.66119827871442481374467622682, 5.36244298721167160718725340278, 6.05886413503827459123865903257, 6.37869009161194820234692371426, 6.79533660929922109224513153399, 7.22142790948264379459326388401, 7.972624440390951709752181868622, 8.000937863938982294341146593643, 9.080746654144001493159861980940