L(s) = 1 | − 2·3-s + 4·5-s − 2·7-s + 2·9-s − 6·13-s − 8·15-s − 6·17-s − 12·19-s + 4·21-s − 6·23-s + 11·25-s − 6·27-s − 8·35-s − 6·37-s + 12·39-s − 12·41-s − 6·43-s + 8·45-s + 18·47-s + 2·49-s + 12·51-s − 10·53-s + 24·57-s − 20·59-s + 24·61-s − 4·63-s − 24·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.755·7-s + 2/3·9-s − 1.66·13-s − 2.06·15-s − 1.45·17-s − 2.75·19-s + 0.872·21-s − 1.25·23-s + 11/5·25-s − 1.15·27-s − 1.35·35-s − 0.986·37-s + 1.92·39-s − 1.87·41-s − 0.914·43-s + 1.19·45-s + 2.62·47-s + 2/7·49-s + 1.68·51-s − 1.37·53-s + 3.17·57-s − 2.60·59-s + 3.07·61-s − 0.503·63-s − 2.97·65-s + ⋯ |
Λ(s)=(=(1638400s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(1638400s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1638400
= 216⋅52
|
Sign: |
1
|
Analytic conductor: |
104.465 |
Root analytic conductor: |
3.19700 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 1638400, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.606299836933068250284171331592, −9.199410844352709356249726533898, −8.632318355985944144310071390822, −8.573553673398465119919932228310, −7.74162940427514495117161920140, −7.14168588225264393809283129143, −6.74614447926115849110520562244, −6.55790017168689678450898063968, −5.98330834278771211460823296822, −5.97872457856260773200563781721, −5.24034670808618061461712700234, −4.99151588020140837812434783222, −4.23933687504990012576776971278, −4.20491201335763035945327198834, −3.10793775980364593750376803825, −2.46298842795261715058988739993, −1.98380104940191069438811831423, −1.76459039528730533636464057857, 0, 0,
1.76459039528730533636464057857, 1.98380104940191069438811831423, 2.46298842795261715058988739993, 3.10793775980364593750376803825, 4.20491201335763035945327198834, 4.23933687504990012576776971278, 4.99151588020140837812434783222, 5.24034670808618061461712700234, 5.97872457856260773200563781721, 5.98330834278771211460823296822, 6.55790017168689678450898063968, 6.74614447926115849110520562244, 7.14168588225264393809283129143, 7.74162940427514495117161920140, 8.573553673398465119919932228310, 8.632318355985944144310071390822, 9.199410844352709356249726533898, 9.606299836933068250284171331592