L(s) = 1 | − 1.73i·3-s + 6.29·5-s + 2.64i·7-s − 2.99·9-s − 16.5i·11-s − 14.1·13-s − 10.8i·15-s − 20.4·17-s − 6.39i·19-s + 4.58·21-s + 29.0i·23-s + 14.5·25-s + 5.19i·27-s − 54.6·29-s − 14.7i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.25·5-s + 0.377i·7-s − 0.333·9-s − 1.50i·11-s − 1.08·13-s − 0.726i·15-s − 1.20·17-s − 0.336i·19-s + 0.218·21-s + 1.26i·23-s + 0.583·25-s + 0.192i·27-s − 1.88·29-s − 0.475i·31-s + ⋯ |
Λ(s)=(=(1344s/2ΓC(s)L(s)−Λ(3−s)
Λ(s)=(=(1344s/2ΓC(s+1)L(s)−Λ(1−s)
Degree: |
2 |
Conductor: |
1344
= 26⋅3⋅7
|
Sign: |
−1
|
Analytic conductor: |
36.6213 |
Root analytic conductor: |
6.05155 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1344(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1344, ( :1), −1)
|
Particular Values
L(23) |
≈ |
0.6972980165 |
L(21) |
≈ |
0.6972980165 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+1.73iT |
| 7 | 1−2.64iT |
good | 5 | 1−6.29T+25T2 |
| 11 | 1+16.5iT−121T2 |
| 13 | 1+14.1T+169T2 |
| 17 | 1+20.4T+289T2 |
| 19 | 1+6.39iT−361T2 |
| 23 | 1−29.0iT−529T2 |
| 29 | 1+54.6T+841T2 |
| 31 | 1+14.7iT−961T2 |
| 37 | 1+26.8T+1.36e3T2 |
| 41 | 1+33.5T+1.68e3T2 |
| 43 | 1+3.83iT−1.84e3T2 |
| 47 | 1+27.4iT−2.20e3T2 |
| 53 | 1−21.7T+2.80e3T2 |
| 59 | 1−54.1iT−3.48e3T2 |
| 61 | 1−35.7T+3.72e3T2 |
| 67 | 1+95.4iT−4.48e3T2 |
| 71 | 1+55.6iT−5.04e3T2 |
| 73 | 1−130.T+5.32e3T2 |
| 79 | 1−15.6iT−6.24e3T2 |
| 83 | 1+49.4iT−6.88e3T2 |
| 89 | 1−39.2T+7.92e3T2 |
| 97 | 1+171.T+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
−9.166990565096219587258027584737, −8.283190687872264232595832780215, −7.29031376756199167462765353891, −6.46871806662027178777662270397, −5.66588382451877625305889519182, −5.18101182989554213760701990290, −3.59592524127049180421950429163, −2.46463181534381209958310861604, −1.74326019648587161615052525785, −0.17113987515956676400818955415,
1.85004898980956121352778875709, 2.47488133900435770534824268457, 3.99764164683947346150407015579, 4.81493957910421931686239927879, 5.48693742897212465494284078196, 6.65683517283155025328272498895, 7.16072853250710587303704395880, 8.360304089005225183439381203463, 9.385883184492363308586961131566, 9.743018613429851354996227892334