L(s) = 1 | + (1.23 + 2.13i)2-s + (−2.02 + 3.51i)4-s + (−1.82 + 3.16i)5-s − 5.05·8-s − 9.00·10-s + (0.203 + 0.351i)11-s + (0.243 − 0.421i)13-s + (−2.16 − 3.74i)16-s − 4.85·17-s − 1.97·19-s + (−7.41 − 12.8i)20-s + (−0.5 + 0.866i)22-s + (2.32 − 4.02i)23-s + (−4.19 − 7.25i)25-s + 1.19·26-s + ⋯ |
L(s) = 1 | + (0.869 + 1.50i)2-s + (−1.01 + 1.75i)4-s + (−0.817 + 1.41i)5-s − 1.78·8-s − 2.84·10-s + (0.0612 + 0.106i)11-s + (0.0675 − 0.116i)13-s + (−0.540 − 0.936i)16-s − 1.17·17-s − 0.452·19-s + (−1.65 − 2.87i)20-s + (−0.106 + 0.184i)22-s + (0.484 − 0.839i)23-s + (−0.838 − 1.45i)25-s + 0.234·26-s + ⋯ |
Λ(s)=(=(1323s/2ΓC(s)L(s)(−0.0644+0.997i)Λ(2−s)
Λ(s)=(=(1323s/2ΓC(s+1/2)L(s)(−0.0644+0.997i)Λ(1−s)
Degree: |
2 |
Conductor: |
1323
= 33⋅72
|
Sign: |
−0.0644+0.997i
|
Analytic conductor: |
10.5642 |
Root analytic conductor: |
3.25026 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1323(442,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1323, ( :1/2), −0.0644+0.997i)
|
Particular Values
L(1) |
≈ |
1.369619651 |
L(21) |
≈ |
1.369619651 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1+(−1.23−2.13i)T+(−1+1.73i)T2 |
| 5 | 1+(1.82−3.16i)T+(−2.5−4.33i)T2 |
| 11 | 1+(−0.203−0.351i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−0.243+0.421i)T+(−6.5−11.2i)T2 |
| 17 | 1+4.85T+17T2 |
| 19 | 1+1.97T+19T2 |
| 23 | 1+(−2.32+4.02i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−3.82−6.62i)T+(−14.5+25.1i)T2 |
| 31 | 1+(3.51−6.08i)T+(−15.5−26.8i)T2 |
| 37 | 1−2.32T+37T2 |
| 41 | 1+(−3.75+6.50i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−1.16−2.01i)T+(−21.5+37.2i)T2 |
| 47 | 1+(3.15+5.47i)T+(−23.5+40.7i)T2 |
| 53 | 1−3.56T+53T2 |
| 59 | 1+(−3.05+5.29i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−4.01−6.95i)T+(−30.5+52.8i)T2 |
| 67 | 1+(1.80−3.11i)T+(−33.5−58.0i)T2 |
| 71 | 1+8.46T+71T2 |
| 73 | 1−1.97T+73T2 |
| 79 | 1+(4.08+7.06i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−6.08−10.5i)T+(−41.5+71.8i)T2 |
| 89 | 1+14.8T+89T2 |
| 97 | 1+(−4.74−8.21i)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
−10.47499898846932857839595511727, −8.934263840082250891388201377252, −8.346811522525750081785554718805, −7.37688759832109467666402904995, −6.83959913929082937267750909238, −6.44458472128611692848202662911, −5.28762519687642589393601235113, −4.35505954593621784695691165349, −3.61218279556082011461441451257, −2.63556187950415170784297958086,
0.42616091194259896210819977014, 1.58358840935837128510513700338, 2.75110621242514839724020129125, 4.06261680819137130365672879211, 4.32883822940685016548528144318, 5.19969379137110303355046431442, 6.14488014562039262571551375340, 7.58272803927237863260109453982, 8.508475249586462490712775582924, 9.241974903670556134677661175462