L(s) = 1 | + (0.866 − 0.498i)2-s + (0.856 − 0.516i)3-s + (0.502 − 0.864i)4-s + (0.485 − 0.874i)6-s + (−0.213 + 0.976i)7-s + (0.00511 − 0.999i)8-s + (0.467 − 0.884i)9-s + (−0.971 + 0.238i)11-s + (−0.0153 − 0.999i)12-s + (−0.157 − 0.987i)13-s + (0.302 + 0.953i)14-s + (−0.494 − 0.869i)16-s + (0.336 − 0.941i)17-s + (−0.0358 − 0.999i)18-s + (0.985 − 0.168i)19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.498i)2-s + (0.856 − 0.516i)3-s + (0.502 − 0.864i)4-s + (0.485 − 0.874i)6-s + (−0.213 + 0.976i)7-s + (0.00511 − 0.999i)8-s + (0.467 − 0.884i)9-s + (−0.971 + 0.238i)11-s + (−0.0153 − 0.999i)12-s + (−0.157 − 0.987i)13-s + (0.302 + 0.953i)14-s + (−0.494 − 0.869i)16-s + (0.336 − 0.941i)17-s + (−0.0358 − 0.999i)18-s + (0.985 − 0.168i)19-s + ⋯ |
Λ(s)=(=(6145s/2ΓR(s)L(s)(−0.997−0.0642i)Λ(1−s)
Λ(s)=(=(6145s/2ΓR(s)L(s)(−0.997−0.0642i)Λ(1−s)
Degree: |
1 |
Conductor: |
6145
= 5⋅1229
|
Sign: |
−0.997−0.0642i
|
Analytic conductor: |
28.5372 |
Root analytic conductor: |
28.5372 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ6145(272,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 6145, (0: ), −0.997−0.0642i)
|
Particular Values
L(21) |
≈ |
0.1022145625−3.177078293i |
L(21) |
≈ |
0.1022145625−3.177078293i |
L(1) |
≈ |
1.559384943−1.242526840i |
L(1) |
≈ |
1.559384943−1.242526840i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 1229 | 1 |
good | 2 | 1+(0.866−0.498i)T |
| 3 | 1+(0.856−0.516i)T |
| 7 | 1+(−0.213+0.976i)T |
| 11 | 1+(−0.971+0.238i)T |
| 13 | 1+(−0.157−0.987i)T |
| 17 | 1+(0.336−0.941i)T |
| 19 | 1+(0.985−0.168i)T |
| 23 | 1+(−0.913−0.407i)T |
| 29 | 1+(−0.820−0.571i)T |
| 31 | 1+(0.881+0.471i)T |
| 37 | 1+(0.705+0.708i)T |
| 41 | 1+(0.228−0.973i)T |
| 43 | 1+(−0.600−0.799i)T |
| 47 | 1+(0.625+0.780i)T |
| 53 | 1+(−0.994+0.107i)T |
| 59 | 1+(−0.596+0.802i)T |
| 61 | 1+(0.767+0.641i)T |
| 67 | 1+(−0.898−0.439i)T |
| 71 | 1+(0.893+0.448i)T |
| 73 | 1+(−0.350−0.936i)T |
| 79 | 1+(0.00511+0.999i)T |
| 83 | 1+(0.690−0.723i)T |
| 89 | 1+(−0.605+0.796i)T |
| 97 | 1+(−0.770−0.637i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−17.88529400674478663969203723470, −16.968378156575721369249372797578, −16.421025140167665080585039192091, −16.03129003495424016779840053664, −15.3499525435746415218594023657, −14.53509247460547810924746550654, −14.19447683215840649911011480733, −13.46070483675899442736650024819, −13.1382656665017760233091488249, −12.31412303095123801966933392283, −11.34435428020692138957083179753, −10.82550844355015419088928816896, −9.93386233956447705027919036334, −9.474411713374449727850720914425, −8.36052900861929293750452873445, −7.84815507420930728217084439204, −7.42251009380563753008368247822, −6.54824131635754690869441892536, −5.75185012639916288474715446895, −4.93926452478877414109335672461, −4.28720696530864709127676641045, −3.678154821530236051595460429025, −3.143746497370113858660870292320, −2.25936014112272806346690102291, −1.45983415131121901928031730520,
0.439907727126570573750693789936, 1.457829173503341991604247321054, 2.42938361343367680079705053775, 2.75625819458420984032811573639, 3.29898113233700086012727134438, 4.299686039398763112160646896003, 5.19117614504858059173610013031, 5.678413920665602396310808301214, 6.44067659524957542549926965974, 7.37969017238794702022078820516, 7.834546081449273187586002883757, 8.692675790950234571410824518811, 9.67144698159782050862714906165, 9.86789793687664400970590153734, 10.818703202561821597208487892135, 11.78615308658147139618921899789, 12.264064324833066269384098913165, 12.69281197234454763605260920638, 13.59582881428072864690166236932, 13.76939331996190308554137292920, 14.7209374723822797818783012540, 15.333940085733216498591521451765, 15.6509794029213124417927382035, 16.334483780611033855507838123476, 17.77329342501459707476914274893