L(s) = 1 | + (−0.930 − 0.365i)3-s + (0.0747 − 0.997i)5-s + (0.733 + 0.680i)9-s + (−0.680 − 0.733i)11-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (0.866 − 0.5i)17-s + (0.930 − 0.365i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.433 − 0.900i)27-s + (0.563 − 0.826i)31-s + (0.365 + 0.930i)33-s + (0.680 − 0.733i)37-s + (−0.149 + 0.988i)39-s + ⋯ |
L(s) = 1 | + (−0.930 − 0.365i)3-s + (0.0747 − 0.997i)5-s + (0.733 + 0.680i)9-s + (−0.680 − 0.733i)11-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (0.866 − 0.5i)17-s + (0.930 − 0.365i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.433 − 0.900i)27-s + (0.563 − 0.826i)31-s + (0.365 + 0.930i)33-s + (0.680 − 0.733i)37-s + (−0.149 + 0.988i)39-s + ⋯ |
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.989+0.146i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.989+0.146i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
−0.989+0.146i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), −0.989+0.146i)
|
Particular Values
L(21) |
≈ |
−0.08292316568−1.123378013i |
L(21) |
≈ |
−0.08292316568−1.123378013i |
L(1) |
≈ |
0.6765205960−0.4252831437i |
L(1) |
≈ |
0.6765205960−0.4252831437i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
| 29 | 1 |
good | 3 | 1+(−0.930−0.365i)T |
| 5 | 1+(0.0747−0.997i)T |
| 11 | 1+(−0.680−0.733i)T |
| 13 | 1+(−0.222−0.974i)T |
| 17 | 1+(0.866−0.5i)T |
| 19 | 1+(0.930−0.365i)T |
| 23 | 1+(−0.826+0.563i)T |
| 31 | 1+(0.563−0.826i)T |
| 37 | 1+(0.680−0.733i)T |
| 41 | 1−iT |
| 43 | 1+(0.433−0.900i)T |
| 47 | 1+(−0.294+0.955i)T |
| 53 | 1+(0.826+0.563i)T |
| 59 | 1+(−0.5−0.866i)T |
| 61 | 1+(−0.149−0.988i)T |
| 67 | 1+(0.955−0.294i)T |
| 71 | 1+(−0.222−0.974i)T |
| 73 | 1+(0.997−0.0747i)T |
| 79 | 1+(0.680−0.733i)T |
| 83 | 1+(0.623−0.781i)T |
| 89 | 1+(0.997+0.0747i)T |
| 97 | 1+(−0.781−0.623i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.55246638786727885658471767857, −21.60978982728990000842395219126, −21.211794947130119090181382544618, −20.099035155740627480015445102684, −19.010100350770703626485110196306, −18.231641024478178773811929191591, −17.8438075446361710951870272070, −16.70771509244754263396711196564, −16.14083239780926132014764617918, −15.138460756264211685543756443159, −14.53313511746232592892307559864, −13.55040756976116501078577398736, −12.348723221069546892900462037789, −11.79670367583180708091433091975, −10.91110524722741955292999615757, −9.97158762391782152560494215923, −9.77150616300949565860811380536, −8.11338878551391375396109082762, −7.16968509899839421931269114085, −6.45006138629895174983192845667, −5.57605715170589922293498039653, −4.60337070258348845471531833483, −3.66662202745057978667205538250, −2.51031599972350954583460934827, −1.27594330761789465533091781795,
0.38043047348351788139841348067, 0.89379853858446677659396697887, 2.25046410711941556611746402327, 3.57989259145117069869207673425, 4.8918733246686262581187565560, 5.469299752157610773806386346573, 6.09073226693838353152703190658, 7.666554116600547102637983139, 7.85618555274279950334042310910, 9.26288743890260242270202880755, 10.082681315822589246588410420519, 11.00084988631390273455265763676, 11.918866224626027871050052378714, 12.50205285819134884148945440827, 13.36030696193714697827200469118, 13.96182077310453090493659916113, 15.58309909022801889859132610143, 15.99317013927629858609631026511, 16.830328108523202885119927340279, 17.55327110555618118136142182599, 18.26720064898220234885123765614, 19.100218265167705867232542660, 20.07815610529323823147865369365, 20.84180907404995391014859991652, 21.66754901743419900437118934830