L(s) = 1 | − 15·2-s − 987·4-s − 1.80e3·5-s + 1.00e4·7-s + 1.07e4·8-s + 2.70e4·10-s − 1.21e5·11-s + 1.42e5·13-s − 1.51e5·14-s + 4.33e5·16-s + 4.95e5·17-s − 8.40e5·19-s + 1.77e6·20-s + 1.82e6·22-s + 5.92e5·23-s − 2.42e6·25-s − 2.14e6·26-s − 9.96e6·28-s − 1.06e7·29-s + 1.28e7·31-s + 2.51e5·32-s − 7.43e6·34-s − 1.82e7·35-s + 7.17e6·37-s + 1.26e7·38-s − 1.94e7·40-s − 9.29e6·41-s + ⋯ |
L(s) = 1 | − 0.662·2-s − 1.92·4-s − 1.29·5-s + 1.58·7-s + 0.929·8-s + 0.855·10-s − 2.50·11-s + 1.38·13-s − 1.05·14-s + 1.65·16-s + 1.43·17-s − 1.48·19-s + 2.48·20-s + 1.66·22-s + 0.441·23-s − 1.24·25-s − 0.919·26-s − 3.06·28-s − 2.80·29-s + 2.50·31-s + 0.0424·32-s − 0.954·34-s − 2.05·35-s + 0.629·37-s + 0.981·38-s − 1.19·40-s − 0.513·41-s + ⋯ |
Λ(s)=(=((310⋅135)s/2ΓC(s)5L(s)−Λ(10−s)
Λ(s)=(=((310⋅135)s/2ΓC(s+9/2)5L(s)−Λ(1−s)
Degree: |
10 |
Conductor: |
310⋅135
|
Sign: |
−1
|
Analytic conductor: |
7.94541×108 |
Root analytic conductor: |
7.76267 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
5
|
Selberg data: |
(10, 310⋅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
−7.34479932770188704101614407882, −7.32048041366611808641467211954, −7.24474218217664709074168206206, −6.43146944984247054531534436405, −6.20712097586951832241324840627, −6.05944702904612976942238715271, −5.73503965023680589402076492967, −5.69805649350041194026837434644, −5.27334035077340168717428421419, −5.07185312068360161165344685006, −4.66760598741898621022299269417, −4.50647220648620523649038054615, −4.33549415577205195066128810935, −4.27100463854319484887376373174, −4.12075583347254689604654219873, −3.26511657795031569860148068458, −3.22366791931610529478036841669, −3.08144236756302089480049040464, −3.07217613623678968420514448092, −2.25134408688810868612373012497, −1.92430050765043199594857755779, −1.79641490188599927354328740065, −1.33503831324259175371003039560, −1.13278098154286283093504415550, −0.946924238943557130341681988103, 0, 0, 0, 0, 0,
0.946924238943557130341681988103, 1.13278098154286283093504415550, 1.33503831324259175371003039560, 1.79641490188599927354328740065, 1.92430050765043199594857755779, 2.25134408688810868612373012497, 3.07217613623678968420514448092, 3.08144236756302089480049040464, 3.22366791931610529478036841669, 3.26511657795031569860148068458, 4.12075583347254689604654219873, 4.27100463854319484887376373174, 4.33549415577205195066128810935, 4.50647220648620523649038054615, 4.66760598741898621022299269417, 5.07185312068360161165344685006, 5.27334035077340168717428421419, 5.69805649350041194026837434644, 5.73503965023680589402076492967, 6.05944702904612976942238715271, 6.20712097586951832241324840627, 6.43146944984247054531534436405, 7.24474218217664709074168206206, 7.32048041366611808641467211954, 7.34479932770188704101614407882