L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 2.19i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−1.09 − 1.89i)10-s + (−3.54 + 2.04i)11-s + (3.53 − 0.706i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.85 − 4.93i)17-s + (6.69 + 3.86i)19-s + (1.89 + 1.09i)20-s + (2.04 − 3.54i)22-s + (−1.23 − 2.14i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.980i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.346 − 0.600i)10-s + (−1.06 + 0.616i)11-s + (0.980 − 0.196i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.691 − 1.19i)17-s + (1.53 + 0.886i)19-s + (0.424 + 0.245i)20-s + (0.435 − 0.755i)22-s + (−0.258 − 0.447i)23-s + ⋯ |
Λ(s)=(=(1638s/2ΓC(s)L(s)(−0.183−0.983i)Λ(2−s)
Λ(s)=(=(1638s/2ΓC(s+1/2)L(s)(−0.183−0.983i)Λ(1−s)
Degree: |
2 |
Conductor: |
1638
= 2⋅32⋅7⋅13
|
Sign: |
−0.183−0.983i
|
Analytic conductor: |
13.0794 |
Root analytic conductor: |
3.61655 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1638(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1638, ( :1/2), −0.183−0.983i)
|
Particular Values
L(1) |
≈ |
1.262193076 |
L(21) |
≈ |
1.262193076 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1 |
| 7 | 1+(−0.866−0.5i)T |
| 13 | 1+(−3.53+0.706i)T |
good | 5 | 1−2.19iT−5T2 |
| 11 | 1+(3.54−2.04i)T+(5.5−9.52i)T2 |
| 17 | 1+(−2.85+4.93i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−6.69−3.86i)T+(9.5+16.4i)T2 |
| 23 | 1+(1.23+2.14i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−4.90−8.50i)T+(−14.5+25.1i)T2 |
| 31 | 1+7.19iT−31T2 |
| 37 | 1+(8.79−5.07i)T+(18.5−32.0i)T2 |
| 41 | 1+(3.14−1.81i)T+(20.5−35.5i)T2 |
| 43 | 1+(1.37−2.38i)T+(−21.5−37.2i)T2 |
| 47 | 1−7.70iT−47T2 |
| 53 | 1−6.31T+53T2 |
| 59 | 1+(−4.20−2.42i)T+(29.5+51.0i)T2 |
| 61 | 1+(−5.63+9.75i)T+(−30.5−52.8i)T2 |
| 67 | 1+(5.49−3.17i)T+(33.5−58.0i)T2 |
| 71 | 1+(7.21+4.16i)T+(35.5+61.4i)T2 |
| 73 | 1−7.37iT−73T2 |
| 79 | 1−3.45T+79T2 |
| 83 | 1+0.332iT−83T2 |
| 89 | 1+(11.7−6.76i)T+(44.5−77.0i)T2 |
| 97 | 1+(−10.0−5.80i)T+(48.5+84.0i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.781317412627081726174908603863, −8.689881531342553825310705619965, −7.904955142515421507871672871517, −7.33254350804985506683332565134, −6.59352887271357040048225028198, −5.56793445149107358329302173078, −4.95228614775219841416014050306, −3.37852648310366344705700059731, −2.64710374567470565514554430844, −1.24957970007362354644070963289,
0.69046869594014072079153430203, 1.65395991825429234863099151121, 3.06286968413610492029204148717, 3.95519820430291287159696866576, 5.15355392670660067505202292672, 5.71812305370841430985424856668, 6.97541309101079898640251358717, 7.85562987640495348077697240584, 8.576380930255167629182561114303, 8.865737608389913135183311639667