L(s) = 1 | + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
Λ(s)=(=(5531904s/2ΓC(s)2L(s)Λ(1−s)
Λ(s)=(=(5531904s/2ΓC(s)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
5531904
= 28⋅32⋅74
|
Sign: |
1
|
Analytic conductor: |
1.37780 |
Root analytic conductor: |
1.08342 |
Motivic weight: |
0 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 5531904, ( :0,0), 1)
|
Particular Values
L(21) |
≈ |
1.682731761 |
L(21) |
≈ |
1.682731761 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C2 | 1−T+T2 |
| 7 | | 1 |
good | 5 | C22 | 1−T2+T4 |
| 11 | C22 | 1−T2+T4 |
| 13 | C1×C1 | (1−T)2(1+T)2 |
| 17 | C22 | 1−T2+T4 |
| 19 | C2 | (1+T+T2)2 |
| 23 | C22 | 1−T2+T4 |
| 29 | C1×C1 | (1−T)2(1+T)2 |
| 31 | C2 | (1−T+T2)2 |
| 37 | C2 | (1−T+T2)2 |
| 41 | C2 | (1+T2)2 |
| 43 | C1×C1 | (1−T)2(1+T)2 |
| 47 | C2 | (1−T+T2)(1+T+T2) |
| 53 | C2 | (1−T+T2)(1+T+T2) |
| 59 | C2 | (1−T+T2)(1+T+T2) |
| 61 | C2 | (1−T+T2)(1+T+T2) |
| 67 | C2 | (1−T+T2)(1+T+T2) |
| 71 | C2 | (1+T2)2 |
| 73 | C2 | (1−T+T2)(1+T+T2) |
| 79 | C2 | (1−T+T2)(1+T+T2) |
| 83 | C1×C1 | (1−T)2(1+T)2 |
| 89 | C22 | 1−T2+T4 |
| 97 | C1×C1 | (1−T)2(1+T)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
−9.347913553467959116379971356881, −8.829984375487463093238543542460, −8.620886641361845600039444231418, −8.191822468902443548097144435338, −8.059504669325826181063202993505, −7.51604790775848753725465989904, −7.18155410114895145859927380363, −6.51005196797831603293376521922, −6.28431738649007638057939817588, −6.12154568390454536581049972812, −5.37298834629383443464695512859, −4.92489926098771545132311834879, −4.44880349665391334457367252945, −4.12244356078066379432436476872, −3.73112614678026613213519520011, −2.85487618404965542030039888056, −2.84971420831768090795136390045, −2.29856121296980983520458619846, −1.71918953262364424806938255793, −0.845557552588317244927063311378,
0.845557552588317244927063311378, 1.71918953262364424806938255793, 2.29856121296980983520458619846, 2.84971420831768090795136390045, 2.85487618404965542030039888056, 3.73112614678026613213519520011, 4.12244356078066379432436476872, 4.44880349665391334457367252945, 4.92489926098771545132311834879, 5.37298834629383443464695512859, 6.12154568390454536581049972812, 6.28431738649007638057939817588, 6.51005196797831603293376521922, 7.18155410114895145859927380363, 7.51604790775848753725465989904, 8.059504669325826181063202993505, 8.191822468902443548097144435338, 8.620886641361845600039444231418, 8.829984375487463093238543542460, 9.347913553467959116379971356881