L(s) = 1 | + (−0.880 + 0.474i)2-s + (−0.222 − 0.974i)3-s + (0.548 − 0.835i)4-s + (−0.819 + 0.572i)5-s + (0.658 + 0.752i)6-s + (−0.763 + 0.645i)7-s + (−0.0862 + 0.996i)8-s + (−0.900 + 0.433i)9-s + (0.449 − 0.893i)10-s + (−0.996 + 0.0804i)11-s + (−0.937 − 0.349i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.740 + 0.671i)15-s + (−0.397 − 0.917i)16-s + (−0.778 − 0.627i)17-s + ⋯ |
L(s) = 1 | + (−0.880 + 0.474i)2-s + (−0.222 − 0.974i)3-s + (0.548 − 0.835i)4-s + (−0.819 + 0.572i)5-s + (0.658 + 0.752i)6-s + (−0.763 + 0.645i)7-s + (−0.0862 + 0.996i)8-s + (−0.900 + 0.433i)9-s + (0.449 − 0.893i)10-s + (−0.996 + 0.0804i)11-s + (−0.937 − 0.349i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.740 + 0.671i)15-s + (−0.397 − 0.917i)16-s + (−0.778 − 0.627i)17-s + ⋯ |
Λ(s)=(=(547s/2ΓR(s)L(s)(0.956−0.290i)Λ(1−s)
Λ(s)=(=(547s/2ΓR(s)L(s)(0.956−0.290i)Λ(1−s)
Degree: |
1 |
Conductor: |
547
|
Sign: |
0.956−0.290i
|
Analytic conductor: |
2.54025 |
Root analytic conductor: |
2.54025 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ547(19,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 547, (0: ), 0.956−0.290i)
|
Particular Values
L(21) |
≈ |
0.3372707509−0.05007374500i |
L(21) |
≈ |
0.3372707509−0.05007374500i |
L(1) |
≈ |
0.4268513019+0.003786822019i |
L(1) |
≈ |
0.4268513019+0.003786822019i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 547 | 1 |
good | 2 | 1+(−0.880+0.474i)T |
| 3 | 1+(−0.222−0.974i)T |
| 5 | 1+(−0.819+0.572i)T |
| 7 | 1+(−0.763+0.645i)T |
| 11 | 1+(−0.996+0.0804i)T |
| 13 | 1+(−0.988+0.149i)T |
| 17 | 1+(−0.778−0.627i)T |
| 19 | 1+(0.756−0.654i)T |
| 23 | 1+(−0.890+0.454i)T |
| 29 | 1+(0.962−0.272i)T |
| 31 | 1+(−0.650−0.759i)T |
| 37 | 1+(0.605+0.795i)T |
| 41 | 1+(−0.5+0.866i)T |
| 43 | 1+(0.874+0.484i)T |
| 47 | 1+(0.948+0.316i)T |
| 53 | 1+(0.211−0.977i)T |
| 59 | 1+(−0.0402+0.999i)T |
| 61 | 1+(0.998−0.0460i)T |
| 67 | 1+(0.166+0.986i)T |
| 71 | 1+(0.529−0.848i)T |
| 73 | 1+(0.740−0.671i)T |
| 79 | 1+(−0.0172−0.999i)T |
| 83 | 1+(−0.632−0.774i)T |
| 89 | 1+(−0.539+0.842i)T |
| 97 | 1+(−0.558−0.829i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−23.34015742181861576491058262193, −22.366217727921959705480945203612, −21.66409615273548659701673732258, −20.61575803394249473727353148466, −20.03758206785540677354054227677, −19.57362819700897267802872230576, −18.40729279286542411496283386346, −17.34251807756756387574417685074, −16.680421811589320360980135792227, −15.86367566568131900284409114011, −15.58124282893195219303800269291, −14.094310061959056971947094066185, −12.649069652190304101053369786714, −12.25157062463682002590635158772, −11.02481738920672104554224918516, −10.400307626171246800524702620866, −9.68460077885569038591804330597, −8.71882843860539296597834806372, −7.87978949734332497535668019048, −6.93483344276089395026828871816, −5.50363759892589540467874746454, −4.27497542954247501643207265326, −3.58716989187106301067019279767, −2.50227783655047562826640070361, −0.58062677563459330181843686783,
0.46164012280288475353094728499, 2.3446122772746657828906715241, 2.81488693718672498802102538476, 4.88833851591345014692614867919, 5.94239118030189219156189207006, 6.84144958468685649925484831179, 7.48606956951288184698791230109, 8.19554030447409992293117320310, 9.32666838922822212766788816536, 10.26124953321356033852424625890, 11.44226826062326982776897930269, 11.87854735894773061712899239611, 13.03990530463796379267089717274, 14.06873350052404683464851446952, 15.15323357300877753460785028743, 15.79860298950657609717695552349, 16.555312271217322292624361227883, 17.86993822523750500085707069051, 18.18228270587855172299969406828, 19.05795676357533605720642932292, 19.631953824879740576650455185235, 20.2858375704271255225521094496, 22.02513453140381099406963271968, 22.666270184744859277996126740348, 23.681581340310064339496889904706