L(s) = 1 | − 161·3-s + 1.80e3·5-s − 1.00e4·7-s − 5.71e3·9-s − 1.21e5·11-s + 1.42e5·13-s − 2.90e5·15-s − 4.95e5·17-s + 8.40e5·19-s + 1.62e6·21-s + 5.92e5·23-s − 2.42e6·25-s + 1.08e6·27-s + 1.06e7·29-s − 1.28e7·31-s + 1.96e7·33-s − 1.82e7·35-s + 7.17e6·37-s − 2.29e7·39-s + 9.29e6·41-s − 1.28e7·43-s − 1.03e7·45-s − 4.33e7·47-s − 3.72e7·49-s + 7.98e7·51-s + 9.32e7·53-s − 2.19e8·55-s + ⋯ |
L(s) = 1 | − 1.14·3-s + 1.29·5-s − 1.58·7-s − 0.290·9-s − 2.50·11-s + 1.38·13-s − 1.48·15-s − 1.43·17-s + 1.48·19-s + 1.82·21-s + 0.441·23-s − 1.24·25-s + 0.392·27-s + 2.80·29-s − 2.50·31-s + 2.87·33-s − 2.05·35-s + 0.629·37-s − 1.59·39-s + 0.513·41-s − 0.572·43-s − 0.374·45-s − 1.29·47-s − 0.923·49-s + 1.65·51-s + 1.62·53-s − 3.23·55-s + ⋯ |
Λ(s)=(=((220⋅135)s/2ΓC(s)5L(s)−Λ(10−s)
Λ(s)=(=((220⋅135)s/2ΓC(s+9/2)5L(s)−Λ(1−s)
Degree: |
10 |
Conductor: |
220⋅135
|
Sign: |
−1
|
Analytic conductor: |
1.41092×1010 |
Root analytic conductor: |
10.3502 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
5
|
Selberg data: |
(10, 220⋅135, ( :9/2,9/2,9/2,9/2,9/2), −1)
|
Particular Values
L(5) |
= |
0 |
L(21) |
= |
0 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏10(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.36462749398838407803608042422, −6.11263202233780018088132064070, −6.11085580338786371832332775452, −6.05247961861548853021674216887, −5.90425844863114287575812582517, −5.33480214382167014845621576701, −5.30524403864015557455964673795, −5.23371043423504310761354920702, −5.08359343174844389881588039583, −4.61478274580712744175322330520, −4.53054225710761360075774885240, −3.99912029286932311295399467493, −3.75829384685768785142009366180, −3.70088849799796462963140336257, −3.38678278814354224273486520182, −2.82386876762059450946972992543, −2.74414690373965449776635639408, −2.72932067970497987323170898050, −2.66579865352240529894891561831, −2.16573680279845600868060409414, −1.73102366334286269743387912110, −1.58865825813432691043483875373, −1.26658690714945075218574020483, −1.13933177173345191038160784256, −0.816540532126358517243907746039, 0, 0, 0, 0, 0,
0.816540532126358517243907746039, 1.13933177173345191038160784256, 1.26658690714945075218574020483, 1.58865825813432691043483875373, 1.73102366334286269743387912110, 2.16573680279845600868060409414, 2.66579865352240529894891561831, 2.72932067970497987323170898050, 2.74414690373965449776635639408, 2.82386876762059450946972992543, 3.38678278814354224273486520182, 3.70088849799796462963140336257, 3.75829384685768785142009366180, 3.99912029286932311295399467493, 4.53054225710761360075774885240, 4.61478274580712744175322330520, 5.08359343174844389881588039583, 5.23371043423504310761354920702, 5.30524403864015557455964673795, 5.33480214382167014845621576701, 5.90425844863114287575812582517, 6.05247961861548853021674216887, 6.11085580338786371832332775452, 6.11263202233780018088132064070, 6.36462749398838407803608042422