L(s) = 1 | + (0.618 + 1.90i)2-s + (−3.92 − 2.85i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (3.00 − 9.23i)6-s + (5.32 − 3.86i)7-s + (−6.47 − 4.70i)8-s + (−1.05 − 3.25i)9-s − 10.0·10-s + (18.0 − 31.7i)11-s + 19.4·12-s + (−18.7 − 57.6i)13-s + (10.6 + 7.73i)14-s + (19.6 − 14.2i)15-s + (4.94 − 15.2i)16-s + (26.5 − 81.8i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.755 − 0.549i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.204 − 0.628i)6-s + (0.287 − 0.208i)7-s + (−0.286 − 0.207i)8-s + (−0.0391 − 0.120i)9-s − 0.316·10-s + (0.493 − 0.869i)11-s + 0.467·12-s + (−0.399 − 1.22i)13-s + (0.203 + 0.147i)14-s + (0.338 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.379 − 1.16i)17-s + ⋯ |
Λ(s)=(=(110s/2ΓC(s)L(s)(0.371+0.928i)Λ(4−s)
Λ(s)=(=(110s/2ΓC(s+3/2)L(s)(0.371+0.928i)Λ(1−s)
Degree: |
2 |
Conductor: |
110
= 2⋅5⋅11
|
Sign: |
0.371+0.928i
|
Analytic conductor: |
6.49021 |
Root analytic conductor: |
2.54758 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ110(71,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 110, ( :3/2), 0.371+0.928i)
|
Particular Values
L(2) |
≈ |
0.794833−0.538071i |
L(21) |
≈ |
0.794833−0.538071i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.618−1.90i)T |
| 5 | 1+(1.54−4.75i)T |
| 11 | 1+(−18.0+31.7i)T |
good | 3 | 1+(3.92+2.85i)T+(8.34+25.6i)T2 |
| 7 | 1+(−5.32+3.86i)T+(105.−326.i)T2 |
| 13 | 1+(18.7+57.6i)T+(−1.77e3+1.29e3i)T2 |
| 17 | 1+(−26.5+81.8i)T+(−3.97e3−2.88e3i)T2 |
| 19 | 1+(−2.21−1.60i)T+(2.11e3+6.52e3i)T2 |
| 23 | 1+70.7T+1.21e4T2 |
| 29 | 1+(2.67−1.94i)T+(7.53e3−2.31e4i)T2 |
| 31 | 1+(−30.9−95.2i)T+(−2.41e4+1.75e4i)T2 |
| 37 | 1+(−264.+191.i)T+(1.56e4−4.81e4i)T2 |
| 41 | 1+(16.6+12.1i)T+(2.12e4+6.55e4i)T2 |
| 43 | 1+469.T+7.95e4T2 |
| 47 | 1+(238.+173.i)T+(3.20e4+9.87e4i)T2 |
| 53 | 1+(121.+373.i)T+(−1.20e5+8.75e4i)T2 |
| 59 | 1+(−477.+347.i)T+(6.34e4−1.95e5i)T2 |
| 61 | 1+(70.0−215.i)T+(−1.83e5−1.33e5i)T2 |
| 67 | 1+177.T+3.00e5T2 |
| 71 | 1+(163.−504.i)T+(−2.89e5−2.10e5i)T2 |
| 73 | 1+(−734.+533.i)T+(1.20e5−3.69e5i)T2 |
| 79 | 1+(−260.−802.i)T+(−3.98e5+2.89e5i)T2 |
| 83 | 1+(−149.+460.i)T+(−4.62e5−3.36e5i)T2 |
| 89 | 1+60.8T+7.04e5T2 |
| 97 | 1+(−79.2−244.i)T+(−7.38e5+5.36e5i)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
−13.01693109184223849865042057452, −11.96107013938567183466456297510, −11.16906217788216769574632096699, −9.778579625802917681215842340039, −8.265416626337633058295149022305, −7.22286596416538320112581942853, −6.20268149792233739495497777644, −5.18690494789197914724195961667, −3.33692212565432408276811844161, −0.54949703419263031430273600822,
1.83811215170196671675891062311, 4.13016955139014695709850161162, 4.93545913443420567199605617826, 6.30155745778655762512825432089, 8.094629874497301186882541504435, 9.465403520533077540722165733843, 10.28252765980396979104157359714, 11.54603408754412595480618450857, 11.97194568006056436837335815168, 13.15134169803468967873329205906