L(s) = 1 | + (0.192 − 1.40i)2-s + (0.881 − 0.471i)3-s + (−1.92 − 0.539i)4-s + (−1.13 − 1.37i)5-s + (−0.490 − 1.32i)6-s + (−3.27 + 2.19i)7-s + (−1.12 + 2.59i)8-s + (0.555 − 0.831i)9-s + (−2.15 + 1.32i)10-s + (−3.58 + 1.08i)11-s + (−1.95 + 0.432i)12-s + (−5.07 − 4.16i)13-s + (2.43 + 5.01i)14-s + (−1.64 − 0.682i)15-s + (3.41 + 2.07i)16-s + (3.69 − 1.53i)17-s + ⋯ |
L(s) = 1 | + (0.136 − 0.990i)2-s + (0.509 − 0.272i)3-s + (−0.962 − 0.269i)4-s + (−0.506 − 0.616i)5-s + (−0.200 − 0.541i)6-s + (−1.23 + 0.827i)7-s + (−0.398 + 0.917i)8-s + (0.185 − 0.277i)9-s + (−0.680 + 0.417i)10-s + (−1.07 + 0.327i)11-s + (−0.563 + 0.124i)12-s + (−1.40 − 1.15i)13-s + (0.651 + 1.34i)14-s + (−0.425 − 0.176i)15-s + (0.854 + 0.519i)16-s + (0.895 − 0.371i)17-s + ⋯ |
Λ(s)=(=(384s/2ΓC(s)L(s)(−0.818−0.574i)Λ(2−s)
Λ(s)=(=(384s/2ΓC(s+1/2)L(s)(−0.818−0.574i)Λ(1−s)
Degree: |
2 |
Conductor: |
384
= 27⋅3
|
Sign: |
−0.818−0.574i
|
Analytic conductor: |
3.06625 |
Root analytic conductor: |
1.75107 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ384(229,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 384, ( :1/2), −0.818−0.574i)
|
Particular Values
L(1) |
≈ |
0.171452+0.542497i |
L(21) |
≈ |
0.171452+0.542497i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.192+1.40i)T |
| 3 | 1+(−0.881+0.471i)T |
good | 5 | 1+(1.13+1.37i)T+(−0.975+4.90i)T2 |
| 7 | 1+(3.27−2.19i)T+(2.67−6.46i)T2 |
| 11 | 1+(3.58−1.08i)T+(9.14−6.11i)T2 |
| 13 | 1+(5.07+4.16i)T+(2.53+12.7i)T2 |
| 17 | 1+(−3.69+1.53i)T+(12.0−12.0i)T2 |
| 19 | 1+(0.395+4.02i)T+(−18.6+3.70i)T2 |
| 23 | 1+(−4.46+0.888i)T+(21.2−8.80i)T2 |
| 29 | 1+(0.526−1.73i)T+(−24.1−16.1i)T2 |
| 31 | 1+(0.107+0.107i)T+31iT2 |
| 37 | 1+(5.97+0.588i)T+(36.2+7.21i)T2 |
| 41 | 1+(0.531+2.67i)T+(−37.8+15.6i)T2 |
| 43 | 1+(−6.31−3.37i)T+(23.8+35.7i)T2 |
| 47 | 1+(2.41+5.83i)T+(−33.2+33.2i)T2 |
| 53 | 1+(−2.32−7.65i)T+(−44.0+29.4i)T2 |
| 59 | 1+(8.94−7.34i)T+(11.5−57.8i)T2 |
| 61 | 1+(7.19+13.4i)T+(−33.8+50.7i)T2 |
| 67 | 1+(1.25+2.35i)T+(−37.2+55.7i)T2 |
| 71 | 1+(−5.92−8.86i)T+(−27.1+65.5i)T2 |
| 73 | 1+(−3.82−2.55i)T+(27.9+67.4i)T2 |
| 79 | 1+(0.938−2.26i)T+(−55.8−55.8i)T2 |
| 83 | 1+(2.92−0.288i)T+(81.4−16.1i)T2 |
| 89 | 1+(−8.98−1.78i)T+(82.2+34.0i)T2 |
| 97 | 1+(11.0+11.0i)T+97iT2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.70313752695327412706123331599, −9.831595208363153520577817997721, −9.168709032539976400432744880168, −8.225711734938810334334981966735, −7.24889198217797676112231272753, −5.54977457186345692652130156026, −4.78215353289513705277311922853, −3.11934621783292926750177981587, −2.60296093220267965987098492295, −0.32608623990050110758379766751,
3.06722276079270279169897339951, 3.85766548325837938157444263631, 5.08640599647340777159540917122, 6.41665783009069042489063260087, 7.34653983101406760872061360944, 7.77143803398566982228744044787, 9.143606262643369894202357612299, 9.911381620975105079984702287298, 10.60989022277253460619907280003, 12.14713342339368874524967209924