L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s + 2·13-s + 2·17-s + 8·23-s + 3·25-s − 4·29-s − 4·35-s + 12·37-s − 12·41-s + 2·43-s − 8·47-s + 6·49-s − 4·55-s − 8·59-s + 20·61-s + 4·65-s + 4·67-s − 4·71-s + 18·73-s + 4·77-s − 12·79-s + 2·83-s + 4·85-s + 16·89-s − 4·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 0.676·35-s + 1.97·37-s − 1.87·41-s + 0.304·43-s − 1.16·47-s + 6/7·49-s − 0.539·55-s − 1.04·59-s + 2.56·61-s + 0.496·65-s + 0.488·67-s − 0.474·71-s + 2.10·73-s + 0.455·77-s − 1.35·79-s + 0.219·83-s + 0.433·85-s + 1.69·89-s − 0.419·91-s + ⋯ |
Λ(s)=(=(15681600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(15681600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
15681600
= 26⋅34⋅52⋅112
|
Sign: |
1
|
Analytic conductor: |
999.872 |
Root analytic conductor: |
5.62323 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 15681600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.090471188 |
L(21) |
≈ |
3.090471188 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C1 | (1−T)2 |
| 11 | C1 | (1+T)2 |
good | 7 | D4 | 1+2T−2T2+2pT3+p2T4 |
| 13 | C4 | 1−2T+10T2−2pT3+p2T4 |
| 17 | D4 | 1−2T+18T2−2pT3+p2T4 |
| 19 | C2 | (1+pT2)2 |
| 23 | C2 | (1−4T+pT2)2 |
| 29 | C2 | (1+2T+pT2)2 |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1−6T+pT2)2 |
| 41 | C2 | (1+6T+pT2)2 |
| 43 | D4 | 1−2T+70T2−2pT3+p2T4 |
| 47 | C2 | (1+4T+pT2)2 |
| 53 | C22 | 1+38T2+p2T4 |
| 59 | C2 | (1+4T+pT2)2 |
| 61 | C2 | (1−10T+pT2)2 |
| 67 | D4 | 1−4T+70T2−4pT3+p2T4 |
| 71 | D4 | 1+4T+78T2+4pT3+p2T4 |
| 73 | D4 | 1−18T+210T2−18pT3+p2T4 |
| 79 | D4 | 1+12T+126T2+12pT3+p2T4 |
| 83 | D4 | 1−2T+14T2−2pT3+p2T4 |
| 89 | D4 | 1−16T+174T2−16pT3+p2T4 |
| 97 | D4 | 1−8T+142T2−8pT3+p2T4 |
show more | | |
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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
−8.681927373356142504717409803471, −8.237266220587868450622337279627, −8.008151874342029604760356293887, −7.52444327848299020739194379237, −6.99534207064231104535886260551, −6.90183151129377935470332399835, −6.33194746375446376072284713970, −6.16702117471608079278945832342, −5.59790009145838515672466575544, −5.41082390874012797406560280836, −4.86482019649560809189618081266, −4.73108754265234899709649356983, −3.81727508144634279978041275167, −3.71360024776516049312484128673, −2.99048750587129398976343581364, −2.89622172001196878056084332535, −2.22995185898318267104759345975, −1.78009553567555265826743142776, −1.07440916665369499545697458968, −0.56887589256857787452940651825,
0.56887589256857787452940651825, 1.07440916665369499545697458968, 1.78009553567555265826743142776, 2.22995185898318267104759345975, 2.89622172001196878056084332535, 2.99048750587129398976343581364, 3.71360024776516049312484128673, 3.81727508144634279978041275167, 4.73108754265234899709649356983, 4.86482019649560809189618081266, 5.41082390874012797406560280836, 5.59790009145838515672466575544, 6.16702117471608079278945832342, 6.33194746375446376072284713970, 6.90183151129377935470332399835, 6.99534207064231104535886260551, 7.52444327848299020739194379237, 8.008151874342029604760356293887, 8.237266220587868450622337279627, 8.681927373356142504717409803471