L(s) = 1 | + (−1.22 − 0.707i)2-s + (1.72 − 0.158i)3-s + (0.999 + 1.73i)4-s + (−2.22 − 1.02i)6-s − 2.82i·8-s + (2.94 − 0.548i)9-s + (−3.27 − 1.89i)11-s + (1.99 + 2.82i)12-s + (−2.00 + 3.46i)16-s + 8.02i·17-s + (−3.99 − 1.41i)18-s − 8.34·19-s + (2.67 + 4.63i)22-s + (−0.449 − 4.87i)24-s + (2.5 − 4.33i)25-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.995 − 0.0917i)3-s + (0.499 + 0.866i)4-s + (−0.908 − 0.418i)6-s − 0.999i·8-s + (0.983 − 0.182i)9-s + (−0.987 − 0.570i)11-s + (0.577 + 0.816i)12-s + (−0.500 + 0.866i)16-s + 1.94i·17-s + (−0.942 − 0.333i)18-s − 1.91·19-s + (0.570 + 0.987i)22-s + (−0.0917 − 0.995i)24-s + (0.5 − 0.866i)25-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(0.870+0.491i)Λ(2−s)
Λ(s)=(=(72s/2ΓC(s+1/2)L(s)(0.870+0.491i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
0.870+0.491i
|
Analytic conductor: |
0.574922 |
Root analytic conductor: |
0.758236 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :1/2), 0.870+0.491i)
|
Particular Values
L(1) |
≈ |
0.797740−0.209813i |
L(21) |
≈ |
0.797740−0.209813i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.22+0.707i)T |
| 3 | 1+(−1.72+0.158i)T |
good | 5 | 1+(−2.5+4.33i)T2 |
| 7 | 1+(3.5+6.06i)T2 |
| 11 | 1+(3.27+1.89i)T+(5.5+9.52i)T2 |
| 13 | 1+(6.5−11.2i)T2 |
| 17 | 1−8.02iT−17T2 |
| 19 | 1+8.34T+19T2 |
| 23 | 1+(−11.5+19.9i)T2 |
| 29 | 1+(−14.5−25.1i)T2 |
| 31 | 1+(15.5−26.8i)T2 |
| 37 | 1−37T2 |
| 41 | 1+(0.398−0.230i)T+(20.5−35.5i)T2 |
| 43 | 1+(−1.17+2.03i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−23.5−40.7i)T2 |
| 53 | 1+53T2 |
| 59 | 1+(−10.6+6.13i)T+(29.5−51.0i)T2 |
| 61 | 1+(30.5+52.8i)T2 |
| 67 | 1+(−7.17−12.4i)T+(−33.5+58.0i)T2 |
| 71 | 1+71T2 |
| 73 | 1−13.6T+73T2 |
| 79 | 1+(39.5+68.4i)T2 |
| 83 | 1+(2.44+1.41i)T+(41.5+71.8i)T2 |
| 89 | 1+5.65iT−89T2 |
| 97 | 1+(9.84−17.0i)T+(−48.5−84.0i)T2 |
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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
−14.68592575309039060882362215601, −13.13253616871827371677520951235, −12.59712718113872897415425981215, −10.82257133950445822006569975190, −10.12341874679207186463613675070, −8.542133080835188017557302649565, −8.193271439978103121820558346488, −6.56921485253876360296646337170, −3.89150805359828389086771997983, −2.26572919897296598094005540027,
2.46497567617430314622651811016, 4.91148107712993364417179104961, 6.88674705314451405041949624189, 7.86177047040072258534126108663, 8.976847986167529653157356874982, 9.918566536362943500092950394430, 11.01267020733973071648887699286, 12.75793173445158789758196750730, 13.94325831746413774485719409405, 14.99026046468207165228230224067