L(s) = 1 | + (−0.365 + 0.930i)5-s + (0.5 − 0.866i)7-s + (−0.900 − 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.365 − 0.930i)17-s + (−0.826 − 0.563i)19-s + (0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (−0.955 − 0.294i)29-s + (0.733 − 0.680i)31-s + (0.623 + 0.781i)35-s + (−0.5 − 0.866i)37-s + (0.222 + 0.974i)41-s + (−0.900 + 0.433i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.365 + 0.930i)5-s + (0.5 − 0.866i)7-s + (−0.900 − 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.365 − 0.930i)17-s + (−0.826 − 0.563i)19-s + (0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (−0.955 − 0.294i)29-s + (0.733 − 0.680i)31-s + (0.623 + 0.781i)35-s + (−0.5 − 0.866i)37-s + (0.222 + 0.974i)41-s + (−0.900 + 0.433i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
Λ(s)=(=(516s/2ΓR(s)L(s)(−0.628−0.778i)Λ(1−s)
Λ(s)=(=(516s/2ΓR(s)L(s)(−0.628−0.778i)Λ(1−s)
Degree: |
1 |
Conductor: |
516
= 22⋅3⋅43
|
Sign: |
−0.628−0.778i
|
Analytic conductor: |
2.39629 |
Root analytic conductor: |
2.39629 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ516(239,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 516, (0: ), −0.628−0.778i)
|
Particular Values
L(21) |
≈ |
0.1993717783−0.4170864063i |
L(21) |
≈ |
0.1993717783−0.4170864063i |
L(1) |
≈ |
0.7485182766−0.08010232113i |
L(1) |
≈ |
0.7485182766−0.08010232113i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 43 | 1 |
good | 5 | 1+(−0.365+0.930i)T |
| 7 | 1+(0.5−0.866i)T |
| 11 | 1+(−0.900−0.433i)T |
| 13 | 1+(−0.988+0.149i)T |
| 17 | 1+(−0.365−0.930i)T |
| 19 | 1+(−0.826−0.563i)T |
| 23 | 1+(0.0747+0.997i)T |
| 29 | 1+(−0.955−0.294i)T |
| 31 | 1+(0.733−0.680i)T |
| 37 | 1+(−0.5−0.866i)T |
| 41 | 1+(0.222+0.974i)T |
| 47 | 1+(−0.900+0.433i)T |
| 53 | 1+(0.988+0.149i)T |
| 59 | 1+(0.623−0.781i)T |
| 61 | 1+(−0.733−0.680i)T |
| 67 | 1+(−0.826−0.563i)T |
| 71 | 1+(0.0747−0.997i)T |
| 73 | 1+(−0.988+0.149i)T |
| 79 | 1+(0.5−0.866i)T |
| 83 | 1+(0.955−0.294i)T |
| 89 | 1+(−0.955+0.294i)T |
| 97 | 1+(−0.900−0.433i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.11141425196242505037982446390, −23.09764382386010361171724825828, −22.1306414065540974877070721987, −21.158590230865146830799325562593, −20.6766583199313966050298823029, −19.6259372950021362906609264342, −18.886097828200764779288598470874, −17.86678235895600734196754952269, −17.08590102964494878441468113673, −16.21924036274742482623976602899, −15.15943894850980344016747524002, −14.8159119879254041427481515381, −13.32441670212814946446403077418, −12.46736237795673893473584230933, −12.0830176940972279977616730434, −10.79736920204948731411835742671, −9.90382666762618973557849217761, −8.63342947979462893906027373446, −8.28528009945114443942262059816, −7.143134904112022586269216525963, −5.756502371666710440502956875583, −4.98011696751539767349578838943, −4.17141101792496785974775127779, −2.60403934983169745180231348562, −1.670791161356940949976919787925,
0.23008442224580908453333993719, 2.11890460786574593904085030398, 3.07304382849209290085259386692, 4.228412947983997219693755674106, 5.15662496860983864321551670079, 6.48886476229162813512135595657, 7.444642600936010205663880819765, 7.88770378764687865296531792924, 9.34910164918478440264736803194, 10.32796701101659069436196112298, 11.095630517359704980477303852950, 11.71387112144642234339970824429, 13.126672691704534303421813587598, 13.81447609493649144629838553401, 14.7460137752745690570188522743, 15.43901310414931901963978450198, 16.46039801862713460074145147830, 17.45562763615674743106083325079, 18.12117003174150863916799452165, 19.14595088974889948602702971491, 19.733459808425852913361432527307, 20.815804012352666249969474704861, 21.56733813691066449700091641005, 22.51306077810880228729254334605, 23.28499155163564909284294834242