L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.53 + 2.66i)5-s + (0.127 − 0.220i)7-s − 0.999·8-s + 3.07·10-s + (1.72 − 2.99i)11-s + (−2.81 − 4.88i)13-s + (−0.127 − 0.220i)14-s + (−0.5 + 0.866i)16-s + 5.01·17-s − 0.904·19-s + (1.53 − 2.66i)20-s + (−1.72 − 2.99i)22-s + (−2.82 − 4.89i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.688 + 1.19i)5-s + (0.0480 − 0.0832i)7-s − 0.353·8-s + 0.973·10-s + (0.520 − 0.901i)11-s + (−0.782 − 1.35i)13-s + (−0.0339 − 0.0588i)14-s + (−0.125 + 0.216i)16-s + 1.21·17-s − 0.207·19-s + (0.344 − 0.596i)20-s + (−0.368 − 0.637i)22-s + (−0.589 − 1.02i)23-s + ⋯ |
Λ(s)=(=(1458s/2ΓC(s)L(s)(0.5+0.866i)Λ(2−s)
Λ(s)=(=(1458s/2ΓC(s+1/2)L(s)(0.5+0.866i)Λ(1−s)
Degree: |
2 |
Conductor: |
1458
= 2⋅36
|
Sign: |
0.5+0.866i
|
Analytic conductor: |
11.6421 |
Root analytic conductor: |
3.41206 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1458(487,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1458, ( :1/2), 0.5+0.866i)
|
Particular Values
L(1) |
≈ |
2.268522033 |
L(21) |
≈ |
2.268522033 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5+0.866i)T |
| 3 | 1 |
good | 5 | 1+(−1.53−2.66i)T+(−2.5+4.33i)T2 |
| 7 | 1+(−0.127+0.220i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−1.72+2.99i)T+(−5.5−9.52i)T2 |
| 13 | 1+(2.81+4.88i)T+(−6.5+11.2i)T2 |
| 17 | 1−5.01T+17T2 |
| 19 | 1+0.904T+19T2 |
| 23 | 1+(2.82+4.89i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−0.883+1.53i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−5.32−9.23i)T+(−15.5+26.8i)T2 |
| 37 | 1−8.83T+37T2 |
| 41 | 1+(−2.85−4.94i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−4.28+7.41i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−6.26+10.8i)T+(−23.5−40.7i)T2 |
| 53 | 1−2.65T+53T2 |
| 59 | 1+(1.85+3.20i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−1.88+3.27i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.79−6.56i)T+(−33.5+58.0i)T2 |
| 71 | 1+8.42T+71T2 |
| 73 | 1+6.99T+73T2 |
| 79 | 1+(6.55−11.3i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−3.46+5.99i)T+(−41.5−71.8i)T2 |
| 89 | 1+6.80T+89T2 |
| 97 | 1+(4.31−7.47i)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
−9.765108458968073389315886171109, −8.644990710497560818884797321098, −7.78115359346841863424083299701, −6.76275478859783144101673745564, −5.97620075871321614912365304132, −5.35774571345324787599632121839, −4.06985730054284845411436447960, −2.99370849820331393747076848592, −2.56168284459775077006599120658, −0.931829821947786026922724150740,
1.31322012294842728329271208419, 2.44394221907385955318552184250, 4.17740060733371813208804250991, 4.54092932061845182690857185012, 5.60314515554228153481570924626, 6.16638970505204570958860022592, 7.31123012067473401153064700364, 7.86902969151416194173426655081, 9.017746678126506841454626902291, 9.487393057378062789012963117077