L(s) = 1 | + 4·3-s + 6·9-s − 4·11-s − 4·17-s + 4·19-s − 6·25-s − 4·27-s − 16·33-s − 12·41-s + 12·43-s + 2·49-s − 16·51-s + 16·57-s + 28·59-s + 20·67-s + 28·73-s − 24·75-s − 37·81-s − 12·83-s − 4·89-s − 4·97-s − 24·99-s − 4·107-s + 4·113-s − 10·121-s − 48·123-s + 127-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·9-s − 1.20·11-s − 0.970·17-s + 0.917·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 1.87·41-s + 1.82·43-s + 2/7·49-s − 2.24·51-s + 2.11·57-s + 3.64·59-s + 2.44·67-s + 3.27·73-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 0.423·89-s − 0.406·97-s − 2.41·99-s − 0.386·107-s + 0.376·113-s − 0.909·121-s − 4.32·123-s + 0.0887·127-s + ⋯ |
Λ(s)=(=(16384s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(16384s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
16384
= 214
|
Sign: |
1
|
Analytic conductor: |
1.04465 |
Root analytic conductor: |
1.01098 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 16384, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.960380722 |
L(21) |
≈ |
1.960380722 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
good | 3 | C2 | (1−2T+pT2)2 |
| 5 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 7 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 11 | C2 | (1+2T+pT2)2 |
| 13 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 17 | C2 | (1+2T+pT2)2 |
| 19 | C2 | (1−2T+pT2)2 |
| 23 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 41 | C2 | (1+6T+pT2)2 |
| 43 | C2 | (1−6T+pT2)2 |
| 47 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 53 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 59 | C2 | (1−14T+pT2)2 |
| 61 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 67 | C2 | (1−10T+pT2)2 |
| 71 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 73 | C2 | (1−14T+pT2)2 |
| 79 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 83 | C2 | (1+6T+pT2)2 |
| 89 | C2 | (1+2T+pT2)2 |
| 97 | C2 | (1+2T+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
−11.15482415732823435746426270865, −10.22941768705266383849925633731, −9.742591354048855107896325616652, −9.385355573475711088007535446099, −8.728992519223720851294830795286, −8.173183686627627779120693205031, −8.098681382024412731658064042949, −7.33738117792255002879146989083, −6.78571919296342801026858814862, −5.64130249043000575312616744005, −5.16542238567969836254694300670, −3.91058908136761919842529082491, −3.56276783709589056726776122484, −2.46608091328452001690059553408, −2.30676446591740616355718333113,
2.30676446591740616355718333113, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 3.91058908136761919842529082491, 5.16542238567969836254694300670, 5.64130249043000575312616744005, 6.78571919296342801026858814862, 7.33738117792255002879146989083, 8.098681382024412731658064042949, 8.173183686627627779120693205031, 8.728992519223720851294830795286, 9.385355573475711088007535446099, 9.742591354048855107896325616652, 10.22941768705266383849925633731, 11.15482415732823435746426270865