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)(0.570+0.820i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(0.570+0.820i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
0.570+0.820i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(571,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), 0.570+0.820i)
|
Particular Values
L(21) |
≈ |
0.9092162723+0.4751268936i |
L(21) |
≈ |
0.9092162723+0.4751268936i |
L(1) |
≈ |
0.7271997337+0.1491178714i |
L(1) |
≈ |
0.7271997337+0.1491178714i |
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
−21.74848430538033189881198131428, −21.3258748548100128762611142600, −20.34842064124179056788044152851, −19.46769088009864298557287291063, −18.622828194719649418933570929286, −17.444086707287169925167284202011, −17.38155686722480467662244358075, −16.4560424077111700703075097301, −15.550656519101396372878527507726, −14.77110246821903190199748392035, −13.281108022490736723007011425232, −12.89812625085174106529311138543, −12.28935339206638426321018047264, −11.25921760880541782167292923881, −10.308188227122811281168227132255, −9.72452804950961592324567668377, −8.45006310788013090336599592606, −7.690230726717482153155870790771, −6.61199952665786015252158606169, −5.758030809307033126595953907765, −4.89002101706901509067183404815, −4.35513120044298673677820735354, −2.54386587215306887242302003593, −1.59037280022670487280624709662, −0.448972607660311696345252289886,
0.58147353997754605919053171910, 2.21341537806063431998575792252, 3.054150646662338030323074807429, 4.42556552827909327922142923449, 5.160205182460617805216706552710, 6.19624931989452940844619507370, 6.83492973148814387328445917361, 7.70452440773628056333191440010, 9.14470907755305503995462636436, 9.8895106059793750021713656111, 10.8752853749813025462565722649, 11.1389458500976679891973996387, 12.25385600731041774231313158955, 13.149362184864268150343609618058, 14.04680867968636687048951502568, 15.02789043286954847149610630736, 15.66354177310871715371853029165, 16.62834738656139842991290662341, 17.35084917536862246831010279403, 18.08968243025296500728029895338, 18.789506396425741980486357220198, 19.53329991488789384129473443646, 21.01084284688300506368154126155, 21.37121778322892083408372912934, 22.21395852540296586558124683584