L(s) = 1 | + 3.26e3·9-s − 1.19e6·17-s + 3.90e6·25-s + 6.86e7·41-s − 8.07e7·49-s + 8.44e8·73-s − 3.76e8·81-s − 2.17e9·89-s − 3.47e9·97-s + 2.29e9·113-s + 3.54e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.90e9·153-s + 157-s + 163-s + 167-s + 2.12e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 0.165·9-s − 3.47·17-s + 2·25-s + 3.79·41-s − 2·49-s + 3.48·73-s − 0.972·81-s − 3.68·89-s − 3.98·97-s + 1.32·113-s + 0.150·121-s − 0.575·153-s + 2·169-s + ⋯ |
Λ(s)=(=(4096s/2ΓC(s)2L(s)Λ(10−s)
Λ(s)=(=(4096s/2ΓC(s+9/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
4096
= 212
|
Sign: |
1
|
Analytic conductor: |
1086.51 |
Root analytic conductor: |
5.74127 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 4096, ( :9/2,9/2), 1)
|
Particular Values
L(5) |
≈ |
1.524857123 |
L(21) |
≈ |
1.524857123 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
good | 3 | C22 | 1−3266T2+p18T4 |
| 5 | C2 | (1−p9T2)2 |
| 7 | C2 | (1+p9T2)2 |
| 11 | C22 | 1−354349618T2+p18T4 |
| 13 | C2 | (1−p9T2)2 |
| 17 | C2 | (1+597510T+p9T2)2 |
| 19 | C22 | 1+335013705758T2+p18T4 |
| 23 | C2 | (1+p9T2)2 |
| 29 | C2 | (1−p9T2)2 |
| 31 | C2 | (1+p9T2)2 |
| 37 | C2 | (1−p9T2)2 |
| 41 | C2 | (1−34306362T+p9T2)2 |
| 43 | C22 | 1+1000250360894414T2+p18T4 |
| 47 | C2 | (1+p9T2)2 |
| 53 | C2 | (1−p9T2)2 |
| 59 | C22 | 1−10228968070290322T2+p18T4 |
| 61 | C2 | (1−p9T2)2 |
| 67 | C22 | 1−50467064407716994T2+p18T4 |
| 71 | C2 | (1+p9T2)2 |
| 73 | C2 | (1−422324930T+p9T2)2 |
| 79 | C2 | (1+p9T2)2 |
| 83 | C22 | 1−352960460737558306T2+p18T4 |
| 89 | C2 | (1+1089849006T+p9T2)2 |
| 97 | C2 | (1+1738254710T+p9T2)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
−13.03053767191208780186927824832, −12.84807370069876657290302759418, −12.44569336412597739285527916548, −11.17959200870430913064695055609, −11.10182810663012250082376202368, −10.83605369447645095706114592366, −9.646495822014850831218031628157, −9.309392875949209074459744187050, −8.647711668414114287852455210837, −8.185301553968613984526440695682, −7.16589091104376074649291661184, −6.71951468053183707218970546615, −6.22661171593006831491305194470, −5.19643130760178539449584535426, −4.43973078790012342746311078659, −4.11444714593756977299677698945, −2.76251509808669956280400975647, −2.37513433198687195154446743481, −1.33148871414915706942241167858, −0.38656767958788163960738348630,
0.38656767958788163960738348630, 1.33148871414915706942241167858, 2.37513433198687195154446743481, 2.76251509808669956280400975647, 4.11444714593756977299677698945, 4.43973078790012342746311078659, 5.19643130760178539449584535426, 6.22661171593006831491305194470, 6.71951468053183707218970546615, 7.16589091104376074649291661184, 8.185301553968613984526440695682, 8.647711668414114287852455210837, 9.309392875949209074459744187050, 9.646495822014850831218031628157, 10.83605369447645095706114592366, 11.10182810663012250082376202368, 11.17959200870430913064695055609, 12.44569336412597739285527916548, 12.84807370069876657290302759418, 13.03053767191208780186927824832