L(s) = 1 | + 3-s + 3·5-s + 3·9-s + 9·13-s + 3·15-s − 3·17-s + 8·19-s − 9·23-s + 5·25-s + 8·27-s − 15·29-s + 8·31-s + 9·39-s + 15·41-s − 21·43-s + 9·45-s + 3·47-s + 2·49-s − 3·51-s − 3·53-s + 8·57-s − 3·59-s + 7·61-s + 27·65-s − 5·67-s − 9·69-s − 9·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.34·5-s + 9-s + 2.49·13-s + 0.774·15-s − 0.727·17-s + 1.83·19-s − 1.87·23-s + 25-s + 1.53·27-s − 2.78·29-s + 1.43·31-s + 1.44·39-s + 2.34·41-s − 3.20·43-s + 1.34·45-s + 0.437·47-s + 2/7·49-s − 0.420·51-s − 0.412·53-s + 1.05·57-s − 0.390·59-s + 0.896·61-s + 3.34·65-s − 0.610·67-s − 1.08·69-s − 1.06·71-s + ⋯ |
Λ(s)=(=(1478656s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(1478656s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1478656
= 212⋅192
|
Sign: |
1
|
Analytic conductor: |
94.2803 |
Root analytic conductor: |
3.11605 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1478656, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
4.740863842 |
L(21) |
≈ |
4.740863842 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 19 | C2 | 1−8T+pT2 |
good | 3 | C22 | 1−T−2T2−pT3+p2T4 |
| 5 | C22 | 1−3T+4T2−3pT3+p2T4 |
| 7 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 11 | C22 | 1−10T2+p2T4 |
| 13 | C2 | (1−7T+pT2)(1−2T+pT2) |
| 17 | C22 | 1+3T−8T2+3pT3+p2T4 |
| 23 | C22 | 1+9T+50T2+9pT3+p2T4 |
| 29 | C22 | 1+15T+104T2+15pT3+p2T4 |
| 31 | C2 | (1−4T+pT2)2 |
| 37 | C2 | (1−pT2)2 |
| 41 | C22 | 1−15T+116T2−15pT3+p2T4 |
| 43 | C2 | (1+8T+pT2)(1+13T+pT2) |
| 47 | C22 | 1−3T+50T2−3pT3+p2T4 |
| 53 | C22 | 1+3T+56T2+3pT3+p2T4 |
| 59 | C22 | 1+3T−50T2+3pT3+p2T4 |
| 61 | C22 | 1−7T−12T2−7pT3+p2T4 |
| 67 | C2 | (1−11T+pT2)(1+16T+pT2) |
| 71 | C22 | 1+9T+10T2+9pT3+p2T4 |
| 73 | C2 | (1−10T+pT2)(1+17T+pT2) |
| 79 | C22 | 1−7T−30T2−7pT3+p2T4 |
| 83 | C22 | 1−154T2+p2T4 |
| 89 | C22 | 1−15T+164T2−15pT3+p2T4 |
| 97 | C22 | 1−15T+172T2−15pT3+p2T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.613737463799651266481014663386, −9.514038535655649652878248809191, −9.428805445326172294936277762038, −8.587057279022980985520003402159, −8.462408401309023849620288562977, −8.032104236143628588569475981047, −7.40107470901311891332661378989, −7.15160090147679139941393715466, −6.47244652223051024373275277617, −6.02990941586390250197137773875, −5.99907616735547383342897559851, −5.39016111202725860819338940540, −4.81410324125552877133106795505, −4.20744429110713390758247401126, −3.75701938541949445237316644520, −3.35553930924184635747774607117, −2.73387895167499935080275026694, −1.90991638080600337907736422450, −1.63786662380192678424973490117, −0.990804334067465316143698404940,
0.990804334067465316143698404940, 1.63786662380192678424973490117, 1.90991638080600337907736422450, 2.73387895167499935080275026694, 3.35553930924184635747774607117, 3.75701938541949445237316644520, 4.20744429110713390758247401126, 4.81410324125552877133106795505, 5.39016111202725860819338940540, 5.99907616735547383342897559851, 6.02990941586390250197137773875, 6.47244652223051024373275277617, 7.15160090147679139941393715466, 7.40107470901311891332661378989, 8.032104236143628588569475981047, 8.462408401309023849620288562977, 8.587057279022980985520003402159, 9.428805445326172294936277762038, 9.514038535655649652878248809191, 9.613737463799651266481014663386