L(s) = 1 | − 2.64·2-s + 3.00·4-s + (2.64 + 4.24i)5-s + 11.2i·7-s + 2.64·8-s + (−7.00 − 11.2i)10-s + 4.24i·11-s − 11.2i·13-s − 29.6i·14-s − 18.9·16-s − 10.5·17-s + 20·19-s + (7.93 + 12.7i)20-s − 11.2i·22-s + 5.29·23-s + ⋯ |
L(s) = 1 | − 1.32·2-s + 0.750·4-s + (0.529 + 0.848i)5-s + 1.60i·7-s + 0.330·8-s + (−0.700 − 1.12i)10-s + 0.385i·11-s − 0.863i·13-s − 2.12i·14-s − 1.18·16-s − 0.622·17-s + 1.05·19-s + (0.396 + 0.636i)20-s − 0.510i·22-s + 0.230·23-s + ⋯ |
Λ(s)=(=(45s/2ΓC(s)L(s)(0.387−0.921i)Λ(3−s)
Λ(s)=(=(45s/2ΓC(s+1)L(s)(0.387−0.921i)Λ(1−s)
Degree: |
2 |
Conductor: |
45
= 32⋅5
|
Sign: |
0.387−0.921i
|
Analytic conductor: |
1.22616 |
Root analytic conductor: |
1.10732 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ45(44,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 45, ( :1), 0.387−0.921i)
|
Particular Values
L(23) |
≈ |
0.504031+0.334956i |
L(21) |
≈ |
0.504031+0.334956i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(−2.64−4.24i)T |
good | 2 | 1+2.64T+4T2 |
| 7 | 1−11.2iT−49T2 |
| 11 | 1−4.24iT−121T2 |
| 13 | 1+11.2iT−169T2 |
| 17 | 1+10.5T+289T2 |
| 19 | 1−20T+361T2 |
| 23 | 1−5.29T+529T2 |
| 29 | 1+8.48iT−841T2 |
| 31 | 1−26T+961T2 |
| 37 | 1+33.6iT−1.36e3T2 |
| 41 | 1+55.1iT−1.68e3T2 |
| 43 | 1+22.4iT−1.84e3T2 |
| 47 | 1−21.1T+2.20e3T2 |
| 53 | 1−84.6T+2.80e3T2 |
| 59 | 1−46.6iT−3.48e3T2 |
| 61 | 1+22T+3.72e3T2 |
| 67 | 1−89.7iT−4.48e3T2 |
| 71 | 1+50.9iT−5.04e3T2 |
| 73 | 1+67.3iT−5.32e3T2 |
| 79 | 1−14T+6.24e3T2 |
| 83 | 1+74.0T+6.88e3T2 |
| 89 | 1−89.0iT−7.92e3T2 |
| 97 | 1+22.4iT−9.40e3T2 |
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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
−15.84101570607985971672634707062, −15.05312385879001427975443678095, −13.56567886244976263444139814032, −12.00062279755064147655709038920, −10.70126289991570952133228059111, −9.633456267795868645512788391654, −8.674943325615016723439917353331, −7.24622060205251586104663343834, −5.63289825751473972467059571519, −2.42513564231247307750226965335,
1.11221971718424207584124412614, 4.53115976407903653408807930647, 6.82984690131631970870083658319, 8.125213132328614323707477937337, 9.336063511002016076021855870919, 10.22482247917731868148818951590, 11.43854506719165007793598433310, 13.33469185669879375445052312189, 13.94532687656695695115507642992, 16.09078003932604584499458470601