L(s) = 1 | + (−1.67 − 2.89i)3-s + (15.9 + 9.21i)5-s + (15.4 − 10.1i)7-s + (7.91 − 13.7i)9-s + (35.6 − 20.5i)11-s + 24.8i·13-s − 61.5i·15-s + (−41.3 + 23.8i)17-s + (30.9 − 53.5i)19-s + (−55.2 − 27.8i)21-s + (64.3 + 37.1i)23-s + (107. + 185. i)25-s − 143.·27-s − 28.8·29-s + (−11.5 − 20.0i)31-s + ⋯ |
L(s) = 1 | + (−0.321 − 0.557i)3-s + (1.42 + 0.824i)5-s + (0.836 − 0.548i)7-s + (0.293 − 0.507i)9-s + (0.976 − 0.563i)11-s + 0.530i·13-s − 1.06i·15-s + (−0.590 + 0.340i)17-s + (0.373 − 0.646i)19-s + (−0.574 − 0.289i)21-s + (0.583 + 0.337i)23-s + (0.858 + 1.48i)25-s − 1.02·27-s − 0.184·29-s + (−0.0670 − 0.116i)31-s + ⋯ |
Λ(s)=(=(448s/2ΓC(s)L(s)(0.829+0.559i)Λ(4−s)
Λ(s)=(=(448s/2ΓC(s+3/2)L(s)(0.829+0.559i)Λ(1−s)
Degree: |
2 |
Conductor: |
448
= 26⋅7
|
Sign: |
0.829+0.559i
|
Analytic conductor: |
26.4328 |
Root analytic conductor: |
5.14128 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ448(383,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 448, ( :3/2), 0.829+0.559i)
|
Particular Values
L(2) |
≈ |
2.732970502 |
L(21) |
≈ |
2.732970502 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(−15.4+10.1i)T |
good | 3 | 1+(1.67+2.89i)T+(−13.5+23.3i)T2 |
| 5 | 1+(−15.9−9.21i)T+(62.5+108.i)T2 |
| 11 | 1+(−35.6+20.5i)T+(665.5−1.15e3i)T2 |
| 13 | 1−24.8iT−2.19e3T2 |
| 17 | 1+(41.3−23.8i)T+(2.45e3−4.25e3i)T2 |
| 19 | 1+(−30.9+53.5i)T+(−3.42e3−5.94e3i)T2 |
| 23 | 1+(−64.3−37.1i)T+(6.08e3+1.05e4i)T2 |
| 29 | 1+28.8T+2.43e4T2 |
| 31 | 1+(11.5+20.0i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+(51.8−89.8i)T+(−2.53e4−4.38e4i)T2 |
| 41 | 1+96.1iT−6.89e4T2 |
| 43 | 1−195.iT−7.95e4T2 |
| 47 | 1+(89.8−155.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1+(−218.−378.i)T+(−7.44e4+1.28e5i)T2 |
| 59 | 1+(286.+496.i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(368.+212.i)T+(1.13e5+1.96e5i)T2 |
| 67 | 1+(−728.+420.i)T+(1.50e5−2.60e5i)T2 |
| 71 | 1+1.17e3iT−3.57e5T2 |
| 73 | 1+(−716.+413.i)T+(1.94e5−3.36e5i)T2 |
| 79 | 1+(−282.−162.i)T+(2.46e5+4.26e5i)T2 |
| 83 | 1−507.T+5.71e5T2 |
| 89 | 1+(176.+102.i)T+(3.52e5+6.10e5i)T2 |
| 97 | 1−1.11e3iT−9.12e5T2 |
show more | |
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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
−10.83211683569265927328190891247, −9.607361810493884810810848624568, −9.046626896794901301285252468877, −7.57822881980022749248942106901, −6.59745025856543775362442157801, −6.26694795829773359459319187491, −4.95144486173400651195138891766, −3.55729119740226090620351617935, −2.01330997001616826219917375973, −1.11055033503310064053121723260,
1.33726617737530191491384893122, 2.25369887337802688296658388721, 4.24274946909559392166941582175, 5.16453941407837586293930672524, 5.65215586786970183513928278797, 6.94265859108976052931682592236, 8.291074726339640865757636828081, 9.143052287967858577728112416318, 9.804103274756138252995321825677, 10.62114316271152389263581725899