L(s) = 1 | + (4.5 + 7.79i)3-s + (−11.0 + 19.1i)5-s + (−126. + 26.6i)7-s + (−40.5 + 70.1i)9-s + (208. + 360. i)11-s + 797.·13-s − 198.·15-s + (687. + 1.19e3i)17-s + (1.15e3 − 2.00e3i)19-s + (−778. − 869. i)21-s + (−477. + 827. i)23-s + (1.31e3 + 2.28e3i)25-s − 729·27-s − 7.03e3·29-s + (630. + 1.09e3i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.197 + 0.341i)5-s + (−0.978 + 0.205i)7-s + (−0.166 + 0.288i)9-s + (0.519 + 0.899i)11-s + 1.30·13-s − 0.227·15-s + (0.577 + 0.999i)17-s + (0.734 − 1.27i)19-s + (−0.385 − 0.430i)21-s + (−0.188 + 0.326i)23-s + (0.422 + 0.731i)25-s − 0.192·27-s − 1.55·29-s + (0.117 + 0.204i)31-s + ⋯ |
Λ(s)=(=(336s/2ΓC(s)L(s)(−0.968−0.250i)Λ(6−s)
Λ(s)=(=(336s/2ΓC(s+5/2)L(s)(−0.968−0.250i)Λ(1−s)
Degree: |
2 |
Conductor: |
336
= 24⋅3⋅7
|
Sign: |
−0.968−0.250i
|
Analytic conductor: |
53.8889 |
Root analytic conductor: |
7.34091 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ336(289,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 336, ( :5/2), −0.968−0.250i)
|
Particular Values
L(3) |
≈ |
1.316998916 |
L(21) |
≈ |
1.316998916 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(−4.5−7.79i)T |
| 7 | 1+(126.−26.6i)T |
good | 5 | 1+(11.0−19.1i)T+(−1.56e3−2.70e3i)T2 |
| 11 | 1+(−208.−360.i)T+(−8.05e4+1.39e5i)T2 |
| 13 | 1−797.T+3.71e5T2 |
| 17 | 1+(−687.−1.19e3i)T+(−7.09e5+1.22e6i)T2 |
| 19 | 1+(−1.15e3+2.00e3i)T+(−1.23e6−2.14e6i)T2 |
| 23 | 1+(477.−827.i)T+(−3.21e6−5.57e6i)T2 |
| 29 | 1+7.03e3T+2.05e7T2 |
| 31 | 1+(−630.−1.09e3i)T+(−1.43e7+2.47e7i)T2 |
| 37 | 1+(4.88e3−8.46e3i)T+(−3.46e7−6.00e7i)T2 |
| 41 | 1+5.40e3T+1.15e8T2 |
| 43 | 1+1.96e4T+1.47e8T2 |
| 47 | 1+(−1.02e3+1.78e3i)T+(−1.14e8−1.98e8i)T2 |
| 53 | 1+(9.01e3+1.56e4i)T+(−2.09e8+3.62e8i)T2 |
| 59 | 1+(−3.71e3−6.43e3i)T+(−3.57e8+6.19e8i)T2 |
| 61 | 1+(1.74e3−3.02e3i)T+(−4.22e8−7.31e8i)T2 |
| 67 | 1+(−7.92e3−1.37e4i)T+(−6.75e8+1.16e9i)T2 |
| 71 | 1+5.81e4T+1.80e9T2 |
| 73 | 1+(1.95e4+3.38e4i)T+(−1.03e9+1.79e9i)T2 |
| 79 | 1+(−4.88e3+8.45e3i)T+(−1.53e9−2.66e9i)T2 |
| 83 | 1−7.03e4T+3.93e9T2 |
| 89 | 1+(7.21e4−1.24e5i)T+(−2.79e9−4.83e9i)T2 |
| 97 | 1+7.93e4T+8.58e9T2 |
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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
−11.08292648495588681860366588438, −10.10197503684529526236258207510, −9.362013074810042908633863679676, −8.530678442921327649579601560416, −7.25176614841748258323871493723, −6.39871181117403138415471448182, −5.22573297389418479673743106928, −3.79375999447260143362188845650, −3.18652749002496854757248991718, −1.54572592599928544773981665960,
0.34035693491137297772179044406, 1.40741748889352541167801562551, 3.16981203884020678710840393660, 3.83235626691317496463558393072, 5.60952194428769375511132234178, 6.37797623537009223601277914757, 7.46045476203689725062209569663, 8.455958252020127116657901535127, 9.226383918285151176647271872339, 10.21634098439687661354715587828