L(s) = 1 | + (2.73 − 4.73i)2-s + (−10.9 − 18.9i)4-s + (0.0995 − 0.172i)5-s + (−16.0 − 9.31i)7-s − 75.5·8-s + (−0.543 − 0.941i)10-s + (14.2 + 24.6i)11-s + 32.5·13-s + (−87.7 + 50.2i)14-s + (−118. + 206. i)16-s + (−57.7 − 100. i)17-s + (−10.5 + 18.3i)19-s − 4.34·20-s + 155.·22-s + (46.8 − 81.2i)23-s + ⋯ |
L(s) = 1 | + (0.965 − 1.67i)2-s + (−1.36 − 2.36i)4-s + (0.00890 − 0.0154i)5-s + (−0.864 − 0.503i)7-s − 3.33·8-s + (−0.0171 − 0.0297i)10-s + (0.389 + 0.674i)11-s + 0.695·13-s + (−1.67 + 0.959i)14-s + (−1.85 + 3.21i)16-s + (−0.823 − 1.42i)17-s + (−0.127 + 0.221i)19-s − 0.0486·20-s + 1.50·22-s + (0.425 − 0.736i)23-s + ⋯ |
Λ(s)=(=(189s/2ΓC(s)L(s)(−0.550−0.834i)Λ(4−s)
Λ(s)=(=(189s/2ΓC(s+3/2)L(s)(−0.550−0.834i)Λ(1−s)
Degree: |
2 |
Conductor: |
189
= 33⋅7
|
Sign: |
−0.550−0.834i
|
Analytic conductor: |
11.1513 |
Root analytic conductor: |
3.33936 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ189(163,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 189, ( :3/2), −0.550−0.834i)
|
Particular Values
L(2) |
≈ |
0.801352+1.48915i |
L(21) |
≈ |
0.801352+1.48915i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(16.0+9.31i)T |
good | 2 | 1+(−2.73+4.73i)T+(−4−6.92i)T2 |
| 5 | 1+(−0.0995+0.172i)T+(−62.5−108.i)T2 |
| 11 | 1+(−14.2−24.6i)T+(−665.5+1.15e3i)T2 |
| 13 | 1−32.5T+2.19e3T2 |
| 17 | 1+(57.7+100.i)T+(−2.45e3+4.25e3i)T2 |
| 19 | 1+(10.5−18.3i)T+(−3.42e3−5.94e3i)T2 |
| 23 | 1+(−46.8+81.2i)T+(−6.08e3−1.05e4i)T2 |
| 29 | 1+231.T+2.43e4T2 |
| 31 | 1+(140.+243.i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+(−73.2+126.i)T+(−2.53e4−4.38e4i)T2 |
| 41 | 1−111.T+6.89e4T2 |
| 43 | 1−392.T+7.95e4T2 |
| 47 | 1+(−136.+236.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1+(170.+294.i)T+(−7.44e4+1.28e5i)T2 |
| 59 | 1+(348.+603.i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(−185.+320.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(43.5+75.4i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1−88.3T+3.57e5T2 |
| 73 | 1+(−401.−695.i)T+(−1.94e5+3.36e5i)T2 |
| 79 | 1+(182.−315.i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1+921.T+5.71e5T2 |
| 89 | 1+(105.−183.i)T+(−3.52e5−6.10e5i)T2 |
| 97 | 1−845.T+9.12e5T2 |
show more | |
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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
−11.33985291212287500990024800494, −10.91214088761254993040081533127, −9.594739068051937644338758278284, −9.225184162316953500330907099064, −6.99595800971150838431285950785, −5.72086715775267218234658736742, −4.42673692821513344861657515633, −3.52217803080998295577186959819, −2.20020838318055760737312767104, −0.54102078373035419820936887699,
3.22755874584627103530527762770, 4.23528249259268308258170877289, 5.77382350189906130637586280122, 6.24497476392337046211159272760, 7.31376910940433603597690491136, 8.629163887843692456864901078547, 9.090915533305979560240234662751, 10.98097827261234000268232195051, 12.38937996825872917402450390221, 12.99845167477971908771356115500