L(s) = 1 | + 2.38·2-s + 3.69·4-s + (1.46 − 2.52i)5-s + 4.05·8-s + (3.48 − 6.03i)10-s + (−0.676 − 1.17i)11-s + (0.733 + 1.26i)13-s + 2.27·16-s + (1.65 − 2.86i)17-s + (1.10 + 1.91i)19-s + (5.39 − 9.35i)20-s + (−1.61 − 2.79i)22-s + (1.31 − 2.27i)23-s + (−1.76 − 3.05i)25-s + (1.74 + 3.03i)26-s + ⋯ |
L(s) = 1 | + 1.68·2-s + 1.84·4-s + (0.653 − 1.13i)5-s + 1.43·8-s + (1.10 − 1.90i)10-s + (−0.204 − 0.353i)11-s + (0.203 + 0.352i)13-s + 0.568·16-s + (0.401 − 0.695i)17-s + (0.253 + 0.438i)19-s + (1.20 − 2.09i)20-s + (−0.344 − 0.596i)22-s + (0.274 − 0.474i)23-s + (−0.353 − 0.611i)25-s + (0.343 + 0.594i)26-s + ⋯ |
Λ(s)=(=(1323s/2ΓC(s)L(s)(0.747+0.663i)Λ(2−s)
Λ(s)=(=(1323s/2ΓC(s+1/2)L(s)(0.747+0.663i)Λ(1−s)
Degree: |
2 |
Conductor: |
1323
= 33⋅72
|
Sign: |
0.747+0.663i
|
Analytic conductor: |
10.5642 |
Root analytic conductor: |
3.25026 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1323(802,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1323, ( :1/2), 0.747+0.663i)
|
Particular Values
L(1) |
≈ |
4.890322082 |
L(21) |
≈ |
4.890322082 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1−2.38T+2T2 |
| 5 | 1+(−1.46+2.52i)T+(−2.5−4.33i)T2 |
| 11 | 1+(0.676+1.17i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−0.733−1.26i)T+(−6.5+11.2i)T2 |
| 17 | 1+(−1.65+2.86i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.10−1.91i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−1.31+2.27i)T+(−11.5−19.9i)T2 |
| 29 | 1+(0.521−0.903i)T+(−14.5−25.1i)T2 |
| 31 | 1+3.27T+31T2 |
| 37 | 1+(−5.43−9.41i)T+(−18.5+32.0i)T2 |
| 41 | 1+(0.904+1.56i)T+(−20.5+35.5i)T2 |
| 43 | 1+(2.17−3.76i)T+(−21.5−37.2i)T2 |
| 47 | 1−3.97T+47T2 |
| 53 | 1+(−3.22+5.59i)T+(−26.5−45.8i)T2 |
| 59 | 1+12.2T+59T2 |
| 61 | 1+0.559T+61T2 |
| 67 | 1−12.8T+67T2 |
| 71 | 1+12.9T+71T2 |
| 73 | 1+(5.22−9.05i)T+(−36.5−63.2i)T2 |
| 79 | 1−0.767T+79T2 |
| 83 | 1+(0.983−1.70i)T+(−41.5−71.8i)T2 |
| 89 | 1+(−3.20−5.54i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−4.14+7.17i)T+(−48.5−84.0i)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
−9.523548685908559441368909017227, −8.773733762021038356731958087305, −7.74673582163758599042392524634, −6.67043293262754569070531346060, −5.89677627513529099639385408149, −5.18249648802722584640882290212, −4.63658189273941780309465205325, −3.59080528112370906766423374274, −2.57344883179791376718157268520, −1.32237887330186119731725375770,
1.94684372126581882324238709076, 2.85563572801393743926693329118, 3.60082746443330472463175323661, 4.61132549685255244998003001976, 5.70437655198526131688388929002, 6.03040032010998208769594174772, 7.04572978487890701765045246506, 7.58436850993085972315042509610, 9.021346524766586259315363657894, 10.03419549722815090238794609310