L(s) = 1 | − 3i·3-s − 7i·7-s − 9·9-s + 6.11·11-s − 15.2i·13-s − 101. i·17-s + 72.9·19-s − 21·21-s − 138. i·23-s + 27i·27-s − 71.7·29-s + 212.·31-s − 18.3i·33-s + 66.9i·37-s − 45.6·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.167·11-s − 0.324i·13-s − 1.44i·17-s + 0.880·19-s − 0.218·21-s − 1.25i·23-s + 0.192i·27-s − 0.459·29-s + 1.23·31-s − 0.0968i·33-s + 0.297i·37-s − 0.187·39-s + ⋯ |
Λ(s)=(=(2100s/2ΓC(s)L(s)(−0.894+0.447i)Λ(4−s)
Λ(s)=(=(2100s/2ΓC(s+3/2)L(s)(−0.894+0.447i)Λ(1−s)
Degree: |
2 |
Conductor: |
2100
= 22⋅3⋅52⋅7
|
Sign: |
−0.894+0.447i
|
Analytic conductor: |
123.904 |
Root analytic conductor: |
11.1312 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2100(1849,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2100, ( :3/2), −0.894+0.447i)
|
Particular Values
L(2) |
≈ |
1.650695555 |
L(21) |
≈ |
1.650695555 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+3iT |
| 5 | 1 |
| 7 | 1+7iT |
good | 11 | 1−6.11T+1.33e3T2 |
| 13 | 1+15.2iT−2.19e3T2 |
| 17 | 1+101.iT−4.91e3T2 |
| 19 | 1−72.9T+6.85e3T2 |
| 23 | 1+138.iT−1.21e4T2 |
| 29 | 1+71.7T+2.43e4T2 |
| 31 | 1−212.T+2.97e4T2 |
| 37 | 1−66.9iT−5.06e4T2 |
| 41 | 1+12.4T+6.89e4T2 |
| 43 | 1−398.iT−7.95e4T2 |
| 47 | 1+176.iT−1.03e5T2 |
| 53 | 1−131.iT−1.48e5T2 |
| 59 | 1−654.T+2.05e5T2 |
| 61 | 1+120.T+2.26e5T2 |
| 67 | 1+310.iT−3.00e5T2 |
| 71 | 1+400.T+3.57e5T2 |
| 73 | 1+243.iT−3.89e5T2 |
| 79 | 1−553.T+4.93e5T2 |
| 83 | 1+756.iT−5.71e5T2 |
| 89 | 1+80.1T+7.04e5T2 |
| 97 | 1+1.10e3iT−9.12e5T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.291873611187854006842095537476, −7.60123775786496653545902087021, −6.91066186221949859099543521203, −6.20220851884038192561221006329, −5.19208010947653784634598832814, −4.45721565717215445250033683782, −3.22664717279421976539576011856, −2.49843957876396792451094506957, −1.18071549634752870099107060319, −0.37036535816253886462649550025,
1.21143135473428404046741468497, 2.31111213033764325647753014869, 3.47374845251050066080306907759, 4.07753635431654939132569353661, 5.18360490582439903557757144565, 5.78621672022298949313648431573, 6.66326967605597784807901054606, 7.62967474167803303135359313629, 8.411345360852994940456180229166, 9.115933993606922661109550739563