L(s) = 1 | + (2.31 − 1.62i)2-s + (−3.80 + 3.53i)3-s + (2.71 − 7.52i)4-s + (−10.8 − 18.8i)5-s + (−3.06 + 14.3i)6-s + (10.5 + 6.09i)7-s + (−5.94 − 21.8i)8-s + (1.99 − 26.9i)9-s + (−55.7 − 25.8i)10-s + (−22.8 − 13.1i)11-s + (16.2 + 38.2i)12-s + (24.0 − 13.8i)13-s + (34.3 − 3.05i)14-s + (107. + 33.2i)15-s + (−49.2 − 40.8i)16-s + 56.3i·17-s + ⋯ |
L(s) = 1 | + (0.818 − 0.574i)2-s + (−0.732 + 0.680i)3-s + (0.339 − 0.940i)4-s + (−0.971 − 1.68i)5-s + (−0.208 + 0.977i)6-s + (0.570 + 0.329i)7-s + (−0.262 − 0.964i)8-s + (0.0739 − 0.997i)9-s + (−1.76 − 0.819i)10-s + (−0.625 − 0.360i)11-s + (0.391 + 0.920i)12-s + (0.512 − 0.295i)13-s + (0.655 − 0.0582i)14-s + (1.85 + 0.572i)15-s + (−0.769 − 0.638i)16-s + 0.804i·17-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(−0.484+0.874i)Λ(4−s)
Λ(s)=(=(72s/2ΓC(s+3/2)L(s)(−0.484+0.874i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
−0.484+0.874i
|
Analytic conductor: |
4.24813 |
Root analytic conductor: |
2.06110 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :3/2), −0.484+0.874i)
|
Particular Values
L(2) |
≈ |
0.734735−1.24753i |
L(21) |
≈ |
0.734735−1.24753i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−2.31+1.62i)T |
| 3 | 1+(3.80−3.53i)T |
good | 5 | 1+(10.8+18.8i)T+(−62.5+108.i)T2 |
| 7 | 1+(−10.5−6.09i)T+(171.5+297.i)T2 |
| 11 | 1+(22.8+13.1i)T+(665.5+1.15e3i)T2 |
| 13 | 1+(−24.0+13.8i)T+(1.09e3−1.90e3i)T2 |
| 17 | 1−56.3iT−4.91e3T2 |
| 19 | 1−80.9T+6.85e3T2 |
| 23 | 1+(−4.32−7.49i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(−43.8+75.8i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+(−174.+100.i)T+(1.48e4−2.57e4i)T2 |
| 37 | 1+53.3iT−5.06e4T2 |
| 41 | 1+(−398.+229.i)T+(3.44e4−5.96e4i)T2 |
| 43 | 1+(105.−181.i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+(−168.+292.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1+84.2T+1.48e5T2 |
| 59 | 1+(86.9−50.2i)T+(1.02e5−1.77e5i)T2 |
| 61 | 1+(342.+197.i)T+(1.13e5+1.96e5i)T2 |
| 67 | 1+(−68.5−118.i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1+766.T+3.57e5T2 |
| 73 | 1−464.T+3.89e5T2 |
| 79 | 1+(−906.−523.i)T+(2.46e5+4.26e5i)T2 |
| 83 | 1+(−1.13e3−654.i)T+(2.85e5+4.95e5i)T2 |
| 89 | 1−403.iT−7.04e5T2 |
| 97 | 1+(29.9−51.9i)T+(−4.56e5−7.90e5i)T2 |
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
−13.45738685451499429765608125333, −12.41399960028852449070794135512, −11.74765631545565727939258174469, −10.83424466824820712595841535374, −9.386291695252423464246470518013, −8.106096927895219994190517536938, −5.74545007841943911182892263006, −4.90518790303534135936857277867, −3.84376773705442147173109237388, −0.842152391028523753694857443841,
2.90708981404567799799520148492, 4.65698878347123824255709660036, 6.29868419958830746409961277059, 7.30604688044857598003691199457, 7.85458313031624519090617835536, 10.66067384377548549326559022109, 11.40221655955405469087426617619, 12.18048333725423040135006405716, 13.68983624878213294211424338229, 14.34493088990301493287836249141