L(s) = 1 | + (−1.73 + i)2-s + (−5.00 − 8.66i)3-s + (1.99 − 3.46i)4-s + 13.8i·5-s + (17.3 + 10.0i)6-s + (0.227 + 0.131i)7-s + 7.99i·8-s + (−36.5 + 63.3i)9-s + (−13.8 − 23.9i)10-s + (34.6 − 20.0i)11-s − 40.0·12-s − 0.524·14-s + (119. − 69.1i)15-s + (−8 − 13.8i)16-s + (39.8 − 69.0i)17-s − 146. i·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.963 − 1.66i)3-s + (0.249 − 0.433i)4-s + 1.23i·5-s + (1.17 + 0.680i)6-s + (0.0122 + 0.00708i)7-s + 0.353i·8-s + (−1.35 + 2.34i)9-s + (−0.436 − 0.756i)10-s + (0.949 − 0.548i)11-s − 0.963·12-s − 0.0100·14-s + (2.06 − 1.18i)15-s + (−0.125 − 0.216i)16-s + (0.568 − 0.985i)17-s − 1.91i·18-s + ⋯ |
Λ(s)=(=(338s/2ΓC(s)L(s)(−0.890+0.455i)Λ(4−s)
Λ(s)=(=(338s/2ΓC(s+3/2)L(s)(−0.890+0.455i)Λ(1−s)
Degree: |
2 |
Conductor: |
338
= 2⋅132
|
Sign: |
−0.890+0.455i
|
Analytic conductor: |
19.9426 |
Root analytic conductor: |
4.46571 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ338(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 338, ( :3/2), −0.890+0.455i)
|
Particular Values
L(2) |
≈ |
0.4394913483 |
L(21) |
≈ |
0.4394913483 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.73−i)T |
| 13 | 1 |
good | 3 | 1+(5.00+8.66i)T+(−13.5+23.3i)T2 |
| 5 | 1−13.8iT−125T2 |
| 7 | 1+(−0.227−0.131i)T+(171.5+297.i)T2 |
| 11 | 1+(−34.6+20.0i)T+(665.5−1.15e3i)T2 |
| 17 | 1+(−39.8+69.0i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(19.5+11.2i)T+(3.42e3+5.94e3i)T2 |
| 23 | 1+(32.7+56.7i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(20.0+34.7i)T+(−1.21e4+2.11e4i)T2 |
| 31 | 1−113.iT−2.97e4T2 |
| 37 | 1+(−104.+60.5i)T+(2.53e4−4.38e4i)T2 |
| 41 | 1+(343.−198.i)T+(3.44e4−5.96e4i)T2 |
| 43 | 1+(137.−238.i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+440.iT−1.03e5T2 |
| 53 | 1+615.T+1.48e5T2 |
| 59 | 1+(−199.−115.i)T+(1.02e5+1.77e5i)T2 |
| 61 | 1+(−54.7+94.8i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(−191.+110.i)T+(1.50e5−2.60e5i)T2 |
| 71 | 1+(279.+161.i)T+(1.78e5+3.09e5i)T2 |
| 73 | 1+323.iT−3.89e5T2 |
| 79 | 1−743.T+4.93e5T2 |
| 83 | 1+539.iT−5.71e5T2 |
| 89 | 1+(1.09e3−632.i)T+(3.52e5−6.10e5i)T2 |
| 97 | 1+(−282.−163.i)T+(4.56e5+7.90e5i)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
−11.01635799674086847190296606771, −9.948538391168158322788713206624, −8.476733556266051172695346698280, −7.59926351366008627032753177833, −6.63237516965823256935895495246, −6.49021824984571591195566272536, −5.25360762134612938104966849889, −2.92057910463522026015217494556, −1.57810466337066613870949053854, −0.24563890090259189560100686482,
1.26260420774190604024092956249, 3.65162488406968872304946793175, 4.40883596566012560284651688499, 5.39820316317703974474113708464, 6.42992785668851320108156162628, 8.191233025805416066331721189072, 9.138428220151445807934707353790, 9.644287002681821397137759400225, 10.44545091719078221729258105401, 11.38935293743301248461499397370