L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.207 − 0.358i)5-s + (−2.62 + 0.358i)7-s − 0.999·8-s + (0.207 − 0.358i)10-s + (0.5 − 0.866i)11-s − 1.17·13-s + (−1.62 − 2.09i)14-s + (−0.5 − 0.866i)16-s + (−1.08 + 1.88i)17-s + (−0.414 − 0.717i)19-s + 0.414·20-s + 0.999·22-s + (−1.62 − 2.80i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0926 − 0.160i)5-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (0.0654 − 0.113i)10-s + (0.150 − 0.261i)11-s − 0.324·13-s + (−0.433 − 0.558i)14-s + (−0.125 − 0.216i)16-s + (−0.263 + 0.456i)17-s + (−0.0950 − 0.164i)19-s + 0.0926·20-s + 0.213·22-s + (−0.338 − 0.585i)23-s + ⋯ |
Λ(s)=(=(1386s/2ΓC(s)L(s)(0.0725+0.997i)Λ(2−s)
Λ(s)=(=(1386s/2ΓC(s+1/2)L(s)(0.0725+0.997i)Λ(1−s)
Degree: |
2 |
Conductor: |
1386
= 2⋅32⋅7⋅11
|
Sign: |
0.0725+0.997i
|
Analytic conductor: |
11.0672 |
Root analytic conductor: |
3.32675 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1386(991,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1386, ( :1/2), 0.0725+0.997i)
|
Particular Values
L(1) |
≈ |
0.6445626573 |
L(21) |
≈ |
0.6445626573 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5−0.866i)T |
| 3 | 1 |
| 7 | 1+(2.62−0.358i)T |
| 11 | 1+(−0.5+0.866i)T |
good | 5 | 1+(0.207+0.358i)T+(−2.5+4.33i)T2 |
| 13 | 1+1.17T+13T2 |
| 17 | 1+(1.08−1.88i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.414+0.717i)T+(−9.5+16.4i)T2 |
| 23 | 1+(1.62+2.80i)T+(−11.5+19.9i)T2 |
| 29 | 1+2.82T+29T2 |
| 31 | 1+(−3.24+5.61i)T+(−15.5−26.8i)T2 |
| 37 | 1+(4.82+8.36i)T+(−18.5+32.0i)T2 |
| 41 | 1+4.65T+41T2 |
| 43 | 1+2.82T+43T2 |
| 47 | 1+(4.62+8.00i)T+(−23.5+40.7i)T2 |
| 53 | 1+(2.58−4.47i)T+(−26.5−45.8i)T2 |
| 59 | 1+(1.82−3.16i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−0.792−1.37i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−6.74+11.6i)T+(−33.5−58.0i)T2 |
| 71 | 1+13.3T+71T2 |
| 73 | 1+(−2.41+4.18i)T+(−36.5−63.2i)T2 |
| 79 | 1+(2.37+4.11i)T+(−39.5+68.4i)T2 |
| 83 | 1−9.82T+83T2 |
| 89 | 1+(−6.24−10.8i)T+(−44.5+77.0i)T2 |
| 97 | 1+10.1T+97T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.220357982060839557940533711501, −8.560369637468743659283574216933, −7.70742617316236299972913544581, −6.72077057347371523663387637081, −6.22618291882989683056977629327, −5.28979939127568708750041287234, −4.27276199434790416563689152513, −3.44163445302693279683894351679, −2.30229617999755516831390922854, −0.22659812491714184598963774322,
1.51528882943935289287348409276, 2.88633330558794700538444226758, 3.51848597231228586258581817514, 4.61817547246575413571789199545, 5.47412752164126205848966644415, 6.56569202647164996088199418031, 7.09764834662011395415197378905, 8.272343244147741913944752118207, 9.241628047966086435298490054665, 9.849666379413090841142916576333