L(s) = 1 | + (−0.5 − 0.866i)2-s + (4.5 − 7.79i)3-s + (15.5 − 26.8i)4-s + (17 + 29.4i)5-s − 9·6-s − 63·8-s + (−40.5 − 70.1i)9-s + (16.9 − 29.4i)10-s + (170 − 294. i)11-s + (−139.5 − 241. i)12-s + 454·13-s + 306·15-s + (−464.5 − 804. i)16-s + (399 − 691. i)17-s + (−40.5 + 70.1i)18-s + (−446 − 772. i)19-s + ⋯ |
L(s) = 1 | + (−0.0883 − 0.153i)2-s + (0.288 − 0.499i)3-s + (0.484 − 0.838i)4-s + (0.304 + 0.526i)5-s − 0.102·6-s − 0.348·8-s + (−0.166 − 0.288i)9-s + (0.0537 − 0.0931i)10-s + (0.423 − 0.733i)11-s + (−0.279 − 0.484i)12-s + 0.745·13-s + 0.351·15-s + (−0.453 − 0.785i)16-s + (0.334 − 0.579i)17-s + (−0.0294 + 0.0510i)18-s + (−0.283 − 0.490i)19-s + ⋯ |
Λ(s)=(=(147s/2ΓC(s)L(s)(−0.386+0.922i)Λ(6−s)
Λ(s)=(=(147s/2ΓC(s+5/2)L(s)(−0.386+0.922i)Λ(1−s)
Degree: |
2 |
Conductor: |
147
= 3⋅72
|
Sign: |
−0.386+0.922i
|
Analytic conductor: |
23.5764 |
Root analytic conductor: |
4.85555 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ147(67,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 147, ( :5/2), −0.386+0.922i)
|
Particular Values
L(3) |
≈ |
2.266184886 |
L(21) |
≈ |
2.266184886 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−4.5+7.79i)T |
| 7 | 1 |
good | 2 | 1+(0.5+0.866i)T+(−16+27.7i)T2 |
| 5 | 1+(−17−29.4i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(−170+294.i)T+(−8.05e4−1.39e5i)T2 |
| 13 | 1−454T+3.71e5T2 |
| 17 | 1+(−399+691.i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(446+772.i)T+(−1.23e6+2.14e6i)T2 |
| 23 | 1+(−1.59e3−2.76e3i)T+(−3.21e6+5.57e6i)T2 |
| 29 | 1+8.24e3T+2.05e7T2 |
| 31 | 1+(−1.24e3+2.16e3i)T+(−1.43e7−2.47e7i)T2 |
| 37 | 1+(4.89e3+8.48e3i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1−1.98e4T+1.15e8T2 |
| 43 | 1+1.72e4T+1.47e8T2 |
| 47 | 1+(4.46e3+7.73e3i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(75−129.i)T+(−2.09e8−3.62e8i)T2 |
| 59 | 1+(−2.11e4+3.67e4i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(7.37e3+1.27e4i)T+(−4.22e8+7.31e8i)T2 |
| 67 | 1+(−838+1.45e3i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1−1.45e4T+1.80e9T2 |
| 73 | 1+(3.91e4−6.78e4i)T+(−1.03e9−1.79e9i)T2 |
| 79 | 1+(−1.13e3−1.96e3i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1+3.77e4T+3.93e9T2 |
| 89 | 1+(−5.86e4−1.01e5i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1−1.00e4T+8.58e9T2 |
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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
−11.46185058634540938349271985442, −11.04683385692800014174231081683, −9.759684604346111563874226656577, −8.859782024463065321493213918077, −7.37102860987291776611890731681, −6.41132485697829752175078152583, −5.49693741508349489769407815924, −3.40301642306557464678778925580, −2.08247390718390313233196119436, −0.76667496425119150520312119182,
1.71804856053153784102608933785, 3.30980044719199808460636837565, 4.45789763170981101844147039679, 5.99840586250065689225929036539, 7.25496542246421484664806699073, 8.429342519160791736982855892701, 9.145710049258621568770070955734, 10.40239734879337093154202076735, 11.46494446111996955188613008636, 12.58533218067522064031375841121