L(s) = 1 | + 2-s + 4-s + 8-s + 9-s + 16-s + 2·17-s + 18-s − 6·25-s + 32-s + 2·34-s + 36-s + 16·47-s + 10·49-s − 6·50-s + 64-s + 2·68-s + 72-s + 81-s + 20·89-s + 16·94-s + 10·98-s − 6·100-s + 32·103-s − 22·121-s + 127-s + 128-s + 131-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 6/5·25-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 2.33·47-s + 10/7·49-s − 0.848·50-s + 1/8·64-s + 0.242·68-s + 0.117·72-s + 1/9·81-s + 2.11·89-s + 1.65·94-s + 1.01·98-s − 3/5·100-s + 3.15·103-s − 2·121-s + 0.0887·127-s + 0.0883·128-s + 0.0873·131-s + ⋯ |
Λ(s)=(=(332928s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(332928s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
332928
= 27⋅32⋅172
|
Sign: |
1
|
Analytic conductor: |
21.2277 |
Root analytic conductor: |
2.14647 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 332928, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.990522735 |
L(21) |
≈ |
2.990522735 |
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×C1 | (1−T)(1+T) |
| 17 | C2 | 1−2T+pT2 |
good | 5 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 7 | C22 | 1−10T2+p2T4 |
| 11 | C2 | (1+pT2)2 |
| 13 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 19 | C2 | (1+pT2)2 |
| 23 | C22 | 1−10T2+p2T4 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C22 | 1+38T2+p2T4 |
| 37 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 41 | C2 | (1−pT2)2 |
| 43 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 47 | C2 | (1−8T+pT2)2 |
| 53 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 59 | C2 | (1+pT2)2 |
| 61 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 67 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 71 | C22 | 1−42T2+p2T4 |
| 73 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 79 | C22 | 1−122T2+p2T4 |
| 83 | C2 | (1−16T+pT2)(1+16T+pT2) |
| 89 | C2 | (1−10T+pT2)2 |
| 97 | C22 | 1−50T2+p2T4 |
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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
−8.839195003001577747571922628762, −8.199397133263483419221021888103, −7.59266086318036482333944944175, −7.47836930577860038386915655621, −6.90369365410745389037013744241, −6.25353947028937099177490903072, −5.90482930646410958798074482335, −5.42848051451252561002388649254, −4.90421879599021250675981877716, −4.23628918591762555005026706755, −3.87002081775942335033180550859, −3.30466060201462864038082665355, −2.49022410853351334524631782583, −1.97065925433832048403434650419, −0.918455563284789218005293309445,
0.918455563284789218005293309445, 1.97065925433832048403434650419, 2.49022410853351334524631782583, 3.30466060201462864038082665355, 3.87002081775942335033180550859, 4.23628918591762555005026706755, 4.90421879599021250675981877716, 5.42848051451252561002388649254, 5.90482930646410958798074482335, 6.25353947028937099177490903072, 6.90369365410745389037013744241, 7.47836930577860038386915655621, 7.59266086318036482333944944175, 8.199397133263483419221021888103, 8.839195003001577747571922628762