L(s) = 1 | + (0.955 + 0.294i)3-s + (0.365 − 0.930i)5-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s + (0.5 − 0.866i)17-s + (0.955 − 0.294i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)27-s + (−0.988 + 0.149i)31-s + (0.955 − 0.294i)33-s + (−0.826 − 0.563i)37-s + (−0.733 − 0.680i)39-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)3-s + (0.365 − 0.930i)5-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s + (0.5 − 0.866i)17-s + (0.955 − 0.294i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)27-s + (−0.988 + 0.149i)31-s + (0.955 − 0.294i)33-s + (−0.826 − 0.563i)37-s + (−0.733 − 0.680i)39-s + ⋯ |
Λ(s)=(=(812s/2ΓR(s+1)L(s)(0.275−0.961i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(0.275−0.961i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
0.275−0.961i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(303,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), 0.275−0.961i)
|
Particular Values
L(21) |
≈ |
2.779256307−2.093602176i |
L(21) |
≈ |
2.779256307−2.093602176i |
L(1) |
≈ |
1.648683206−0.3769454436i |
L(1) |
≈ |
1.648683206−0.3769454436i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
| 29 | 1 |
good | 3 | 1+(0.955+0.294i)T |
| 5 | 1+(0.365−0.930i)T |
| 11 | 1+(0.826−0.563i)T |
| 13 | 1+(−0.900−0.433i)T |
| 17 | 1+(0.5−0.866i)T |
| 19 | 1+(0.955−0.294i)T |
| 23 | 1+(0.988+0.149i)T |
| 31 | 1+(−0.988+0.149i)T |
| 37 | 1+(−0.826−0.563i)T |
| 41 | 1−T |
| 43 | 1+(0.623−0.781i)T |
| 47 | 1+(0.0747+0.997i)T |
| 53 | 1+(−0.988+0.149i)T |
| 59 | 1+(0.5−0.866i)T |
| 61 | 1+(0.733−0.680i)T |
| 67 | 1+(−0.0747+0.997i)T |
| 71 | 1+(0.900+0.433i)T |
| 73 | 1+(−0.365−0.930i)T |
| 79 | 1+(0.826+0.563i)T |
| 83 | 1+(0.222−0.974i)T |
| 89 | 1+(−0.365+0.930i)T |
| 97 | 1+(0.222−0.974i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.12083950625746352617123347028, −21.40412437679448762031087933600, −20.53357814623960111668846406075, −19.65235724943164364496368618908, −19.07685642936322643519720914950, −18.3559953556651450103231002325, −17.478050355937143477770729069, −16.68144199492574638238237610784, −15.34747325585463134316198445169, −14.70022509138032504609082557771, −14.30131846235191895303422088518, −13.39486318600074559160401853499, −12.44077010145782319553011300192, −11.66252378602445312831813453881, −10.42878155669763245259013008530, −9.70056991079299247066070913459, −9.05472104924354424360171822024, −7.859391657757526256784113608298, −7.09057376059585203278588196370, −6.52596520411397194663011676671, −5.20874828892263654349751891516, −3.9107253706552293851204768382, −3.1797801143186443149047399181, −2.15053661285190689974358984509, −1.353516351611056874389201440989,
0.666671374404031631885141554143, 1.69152201370981552138503316596, 2.8701729698195846922377859174, 3.70567542902910131476718992074, 4.90239695481590260526080232811, 5.43177828437899825075864148006, 6.953793648883783708125651071520, 7.734318622810196408116854834188, 8.76445054390579228786895807953, 9.33185942423025809970228803184, 9.93944605381582931250277082893, 11.16617289736356318741327177811, 12.23333349833468103548897547902, 12.95162764258440326895391874600, 13.93474434068898051248368338560, 14.32560018937404542098970313758, 15.44141080741604803866723334336, 16.18912394021805174706166968267, 16.92305646600495801847851295374, 17.76689460024811629868874101961, 18.93786480383430120452359569892, 19.55658678593973801709994587504, 20.44302113909087127849095480960, 20.76960756157092286914388526889, 21.893373726725706233921904817938