L(s) = 1 | + (1.40 + 0.178i)2-s + (0.5 − 0.866i)3-s + (1.93 + 0.502i)4-s + (−3.33 + 1.92i)5-s + (0.856 − 1.12i)6-s + (−1.59 − 2.11i)7-s + (2.62 + 1.05i)8-s + (−0.499 − 0.866i)9-s + (−5.02 + 2.10i)10-s + (−1.17 − 0.681i)11-s + (1.40 − 1.42i)12-s + 0.369i·13-s + (−1.85 − 3.24i)14-s + 3.85i·15-s + (3.49 + 1.94i)16-s + (3.89 + 2.25i)17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.126i)2-s + (0.288 − 0.499i)3-s + (0.967 + 0.251i)4-s + (−1.49 + 0.862i)5-s + (0.349 − 0.459i)6-s + (−0.602 − 0.798i)7-s + (0.928 + 0.371i)8-s + (−0.166 − 0.288i)9-s + (−1.59 + 0.666i)10-s + (−0.355 − 0.205i)11-s + (0.404 − 0.411i)12-s + 0.102i·13-s + (−0.496 − 0.868i)14-s + 0.995i·15-s + (0.873 + 0.486i)16-s + (0.945 + 0.545i)17-s + ⋯ |
Λ(s)=(=(84s/2ΓC(s)L(s)(0.999+0.00458i)Λ(2−s)
Λ(s)=(=(84s/2ΓC(s+1/2)L(s)(0.999+0.00458i)Λ(1−s)
Degree: |
2 |
Conductor: |
84
= 22⋅3⋅7
|
Sign: |
0.999+0.00458i
|
Analytic conductor: |
0.670743 |
Root analytic conductor: |
0.818989 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ84(31,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 84, ( :1/2), 0.999+0.00458i)
|
Particular Values
L(1) |
≈ |
1.42762−0.00327373i |
L(21) |
≈ |
1.42762−0.00327373i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.40−0.178i)T |
| 3 | 1+(−0.5+0.866i)T |
| 7 | 1+(1.59+2.11i)T |
good | 5 | 1+(3.33−1.92i)T+(2.5−4.33i)T2 |
| 11 | 1+(1.17+0.681i)T+(5.5+9.52i)T2 |
| 13 | 1−0.369iT−13T2 |
| 17 | 1+(−3.89−2.25i)T+(8.5+14.7i)T2 |
| 19 | 1+(0.0330+0.0573i)T+(−9.5+16.4i)T2 |
| 23 | 1+(2.77−1.60i)T+(11.5−19.9i)T2 |
| 29 | 1+3.11T+29T2 |
| 31 | 1+(−3.01+5.22i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−2.74−4.75i)T+(−18.5+32.0i)T2 |
| 41 | 1+8.45iT−41T2 |
| 43 | 1−6.30iT−43T2 |
| 47 | 1+(−0.712−1.23i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−1.27+2.20i)T+(−26.5−45.8i)T2 |
| 59 | 1+(1.71−2.97i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−1.23+0.715i)T+(30.5−52.8i)T2 |
| 67 | 1+(8.45+4.88i)T+(33.5+58.0i)T2 |
| 71 | 1+12.9iT−71T2 |
| 73 | 1+(1.56+0.900i)T+(36.5+63.2i)T2 |
| 79 | 1+(10.8−6.24i)T+(39.5−68.4i)T2 |
| 83 | 1+12.2T+83T2 |
| 89 | 1+(−1.11+0.646i)T+(44.5−77.0i)T2 |
| 97 | 1−2.88iT−97T2 |
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
−14.28187199026268535889178496708, −13.28300157135359719440478597787, −12.23298718692485555528492237591, −11.34366869153124028578394985838, −10.29521344393314774193512108334, −7.938408514420699269606951245231, −7.37823022536774871140927769589, −6.21075714808866561365474181661, −4.06951855965350634757947622721, −3.13236453059016307079577970554,
3.12681587570989946224126052022, 4.37563289662944674232084119740, 5.54593622847326613476353750602, 7.40573416985436534431848597744, 8.556552438397003773863888102476, 9.998685068533117067445997779806, 11.46781050029700819510826232776, 12.23270148744364212753321456722, 12.97759281849599256952338717296, 14.43748537386821591332499902798