L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)10-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + 20-s + (−0.173 − 0.984i)22-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)26-s + (−0.939 + 0.342i)29-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)10-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + 20-s + (−0.173 − 0.984i)22-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)26-s + (−0.939 + 0.342i)29-s + ⋯ |
Λ(s)=(=(399s/2ΓR(s+1)L(s)(−0.243−0.970i)Λ(1−s)
Λ(s)=(=(399s/2ΓR(s+1)L(s)(−0.243−0.970i)Λ(1−s)
Degree: |
1 |
Conductor: |
399
= 3⋅7⋅19
|
Sign: |
−0.243−0.970i
|
Analytic conductor: |
42.8785 |
Root analytic conductor: |
42.8785 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ399(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 399, (1: ), −0.243−0.970i)
|
Particular Values
L(21) |
≈ |
1.897083969−2.431003502i |
L(21) |
≈ |
1.897083969−2.431003502i |
L(1) |
≈ |
1.523771768−0.7623433813i |
L(1) |
≈ |
1.523771768−0.7623433813i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 19 | 1 |
good | 2 | 1+(0.766−0.642i)T |
| 5 | 1+(0.173+0.984i)T |
| 11 | 1+(0.5−0.866i)T |
| 13 | 1+(0.173−0.984i)T |
| 17 | 1+(0.173+0.984i)T |
| 23 | 1+(0.939−0.342i)T |
| 29 | 1+(−0.939+0.342i)T |
| 31 | 1+T |
| 37 | 1+(0.5−0.866i)T |
| 41 | 1+(−0.173−0.984i)T |
| 43 | 1+(0.766−0.642i)T |
| 47 | 1+(0.173−0.984i)T |
| 53 | 1+(0.173−0.984i)T |
| 59 | 1+(−0.173−0.984i)T |
| 61 | 1+(0.939−0.342i)T |
| 67 | 1+(−0.766−0.642i)T |
| 71 | 1+(0.766−0.642i)T |
| 73 | 1+(−0.766+0.642i)T |
| 79 | 1+(0.939+0.342i)T |
| 83 | 1+(−0.5+0.866i)T |
| 89 | 1+(−0.766−0.642i)T |
| 97 | 1+(−0.939−0.342i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.483323484206518221554707220320, −23.55800282062283644044265609924, −22.86732739311919317032264888149, −21.91078113727774383475224398637, −20.90992369271579633988758409935, −20.51888837618255052212519163939, −19.32093503824038086305914149812, −18.00078769221188442121111692504, −17.080392996730397112039702590437, −16.523811810266133349789037455903, −15.61020650307502643702110224816, −14.68404094139928320795900677374, −13.72195041943808274681363434444, −13.023521340957214902288773787567, −12.03593861800740249235364535662, −11.43581287289433006653224616369, −9.603353553407017351742405004064, −8.978089689672732559683488878657, −7.79416530889055543237399059160, −6.88483546464292399774312392835, −5.851352382107150107867272812834, −4.73468314071697428025628908411, −4.2264162052675079701789187806, −2.716785432608466913516503172344, −1.32914830768348015437254836934,
0.6768337846858851872849963486, 2.091129705390256824631621715035, 3.2076485881996041504086179373, 3.85453822580741799813794570058, 5.409622930963156037611426732542, 6.12703117486389830469345511472, 7.10395644031210534093585764263, 8.52095425432212813081457025606, 9.77204238578841687598981185910, 10.73422059743056903853802659609, 11.15569471632545097221834810923, 12.363121381031987326708291395852, 13.25519957898194290095700625257, 14.12555630955056254833764701957, 14.88814434067470300438822327657, 15.59192976069930230131732887, 16.94220448629933485968710200602, 18.03767769590822718894516314974, 18.98583736256996458368846652097, 19.47165463060949527895060469391, 20.62921379006470391141749360406, 21.44983618941702005693111444274, 22.23508842321731306760680555771, 22.79409698250779078948194970715, 23.71527180853680704806838223344