L(s) = 1 | + (−0.264 + 0.457i)2-s + (−2.94 + 0.546i)3-s + (1.86 + 3.22i)4-s + (−0.819 + 4.93i)5-s + (0.529 − 1.49i)6-s + (2.39 + 1.38i)7-s − 4.08·8-s + (8.40 − 3.22i)9-s + (−2.04 − 1.67i)10-s + (−7.99 − 4.61i)11-s + (−7.24 − 8.48i)12-s + (11.7 − 6.79i)13-s + (−1.26 + 0.731i)14-s + (−0.275 − 14.9i)15-s + (−6.36 + 11.0i)16-s + 12.2·17-s + ⋯ |
L(s) = 1 | + (−0.132 + 0.228i)2-s + (−0.983 + 0.182i)3-s + (0.465 + 0.805i)4-s + (−0.163 + 0.986i)5-s + (0.0883 − 0.249i)6-s + (0.342 + 0.197i)7-s − 0.510·8-s + (0.933 − 0.357i)9-s + (−0.204 − 0.167i)10-s + (−0.726 − 0.419i)11-s + (−0.603 − 0.707i)12-s + (0.905 − 0.522i)13-s + (−0.0904 + 0.0522i)14-s + (−0.0183 − 0.999i)15-s + (−0.397 + 0.688i)16-s + 0.718·17-s + ⋯ |
Λ(s)=(=(45s/2ΓC(s)L(s)(−0.0268−0.999i)Λ(3−s)
Λ(s)=(=(45s/2ΓC(s+1)L(s)(−0.0268−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
45
= 32⋅5
|
Sign: |
−0.0268−0.999i
|
Analytic conductor: |
1.22616 |
Root analytic conductor: |
1.10732 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ45(29,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 45, ( :1), −0.0268−0.999i)
|
Particular Values
L(23) |
≈ |
0.590075+0.606142i |
L(21) |
≈ |
0.590075+0.606142i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(2.94−0.546i)T |
| 5 | 1+(0.819−4.93i)T |
good | 2 | 1+(0.264−0.457i)T+(−2−3.46i)T2 |
| 7 | 1+(−2.39−1.38i)T+(24.5+42.4i)T2 |
| 11 | 1+(7.99+4.61i)T+(60.5+104.i)T2 |
| 13 | 1+(−11.7+6.79i)T+(84.5−146.i)T2 |
| 17 | 1−12.2T+289T2 |
| 19 | 1−20.2T+361T2 |
| 23 | 1+(−1.18−2.05i)T+(−264.5+458.i)T2 |
| 29 | 1+(−30.2−17.4i)T+(420.5+728.i)T2 |
| 31 | 1+(14.7+25.5i)T+(−480.5+832.i)T2 |
| 37 | 1−64.3iT−1.36e3T2 |
| 41 | 1+(−34.5+19.9i)T+(840.5−1.45e3i)T2 |
| 43 | 1+(58.5+33.7i)T+(924.5+1.60e3i)T2 |
| 47 | 1+(−46.6+80.8i)T+(−1.10e3−1.91e3i)T2 |
| 53 | 1+9.82T+2.80e3T2 |
| 59 | 1+(50.6−29.2i)T+(1.74e3−3.01e3i)T2 |
| 61 | 1+(−7.75+13.4i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−13.4+7.78i)T+(2.24e3−3.88e3i)T2 |
| 71 | 1−53.1iT−5.04e3T2 |
| 73 | 1+23.6iT−5.32e3T2 |
| 79 | 1+(17.2−29.9i)T+(−3.12e3−5.40e3i)T2 |
| 83 | 1+(37.6−65.2i)T+(−3.44e3−5.96e3i)T2 |
| 89 | 1+29.1iT−7.92e3T2 |
| 97 | 1+(−54.0−31.1i)T+(4.70e3+8.14e3i)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
−15.91307288501522323118606985560, −15.22534996302718978577428633576, −13.49712467464394461092781349153, −12.06219080731680356012510172686, −11.25622266843112344467808824360, −10.24415717246914362517933182415, −8.159117300027897720386448816613, −6.97583257645533694488713138437, −5.65137793829069887936179797031, −3.35060139408119728236036008176,
1.23739139284384958614677443159, 4.82334484254527890910471385314, 5.97230338397255632390341613318, 7.62662811044669404273722525300, 9.483223962644938215894795746335, 10.70617838885786134720666726326, 11.68109525603225862474948363979, 12.69353225877421921418533363155, 14.09149608822447752017075200964, 15.81539831347632139107942965936