L(s) = 1 | + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (0.955 + 0.294i)5-s + (0.955 − 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.5 − 0.866i)11-s + (0.955 + 0.294i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (0.955 + 0.294i)5-s + (0.955 − 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.5 − 0.866i)11-s + (0.955 + 0.294i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + ⋯ |
Λ(s)=(=(547s/2ΓR(s)L(s)(0.994+0.107i)Λ(1−s)
Λ(s)=(=(547s/2ΓR(s)L(s)(0.994+0.107i)Λ(1−s)
Degree: |
1 |
Conductor: |
547
|
Sign: |
0.994+0.107i
|
Analytic conductor: |
2.54025 |
Root analytic conductor: |
2.54025 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ547(117,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 547, (0: ), 0.994+0.107i)
|
Particular Values
L(21) |
≈ |
3.190376515+0.1715713218i |
L(21) |
≈ |
3.190376515+0.1715713218i |
L(1) |
≈ |
2.253637072+0.1732705043i |
L(1) |
≈ |
2.253637072+0.1732705043i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 547 | 1 |
good | 2 | 1+(0.826+0.563i)T |
| 3 | 1+(0.623−0.781i)T |
| 5 | 1+(0.955+0.294i)T |
| 7 | 1+(0.365−0.930i)T |
| 11 | 1+(−0.5−0.866i)T |
| 13 | 1+(0.0747+0.997i)T |
| 17 | 1+(−0.733+0.680i)T |
| 19 | 1+(0.826−0.563i)T |
| 23 | 1+(0.826−0.563i)T |
| 29 | 1+(−0.222+0.974i)T |
| 31 | 1+(0.623−0.781i)T |
| 37 | 1+(−0.733+0.680i)T |
| 41 | 1+(−0.5−0.866i)T |
| 43 | 1+(0.365−0.930i)T |
| 47 | 1+(−0.5+0.866i)T |
| 53 | 1+(0.0747+0.997i)T |
| 59 | 1+(−0.5−0.866i)T |
| 61 | 1+(−0.733+0.680i)T |
| 67 | 1+(−0.733+0.680i)T |
| 71 | 1+(−0.733+0.680i)T |
| 73 | 1+(0.826+0.563i)T |
| 79 | 1+(0.623+0.781i)T |
| 83 | 1+(−0.5−0.866i)T |
| 89 | 1+(−0.222+0.974i)T |
| 97 | 1+(0.826−0.563i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.83794270592362211772552438362, −22.517239013141798764461449331876, −21.38887007936881420004646293040, −21.05976274922637340503384812872, −20.36027175330014265656087967006, −19.57388688418723893218810274304, −18.33353334123143744403140589066, −17.69844567778795522270609240404, −16.24063293872172257672993524870, −15.38063900502948316750364763285, −14.93387931550252151426863109729, −13.883560015542754052039677513046, −13.2518097356317221154033612574, −12.36287432070693064974755276563, −11.29536849134795405962338249912, −10.278166569659088833461784015209, −9.66295002369943201487640101765, −8.88086119664684686480052751627, −7.61956690474638981100143158442, −6.10452856600170752647355711282, −5.05276841456149886735678787408, −4.89340897643932032718044292389, −3.265015418395953552439140968094, −2.5197527093108023538261084401, −1.657717446680698740055901126734,
1.4392708154942314780942385956, 2.54092706106449090204585566624, 3.44161804885907100315061330636, 4.62835078507671249517153231194, 5.79625579216885136638218524677, 6.73557194805141998726009040857, 7.22054574990556879101848362406, 8.39201047804382350529420567915, 9.12783716991760636981208798979, 10.64526170555989467324210512183, 11.45165883651249845861325387415, 12.69540871332392257605997935191, 13.582743437655193306126576523978, 13.7854786653385056380988632668, 14.55960796602946213921124901011, 15.555018905996537733623442564860, 16.79575053652030216609922850748, 17.345003496631951335260734541449, 18.26109364628745484031925408041, 19.16091172499221993453213208766, 20.39023573320116925752209111752, 20.89709156151482918407330856961, 21.758664717173912476583899107253, 22.61281914406163326882107903935, 23.82765921589526435787254726850