L(s) = 1 | + 8.29i·2-s + 20.3i·3-s − 36.8·4-s + (55.0 + 9.59i)5-s − 168.·6-s + 101. i·7-s − 39.8i·8-s − 169.·9-s + (−79.5 + 456. i)10-s − 362.·11-s − 747. i·12-s − 540. i·13-s − 845.·14-s + (−194. + 1.11e3i)15-s − 847.·16-s + 561. i·17-s + ⋯ |
L(s) = 1 | + 1.46i·2-s + 1.30i·3-s − 1.15·4-s + (0.985 + 0.171i)5-s − 1.91·6-s + 0.786i·7-s − 0.219i·8-s − 0.698·9-s + (−0.251 + 1.44i)10-s − 0.903·11-s − 1.49i·12-s − 0.886i·13-s − 1.15·14-s + (−0.223 + 1.28i)15-s − 0.827·16-s + 0.471i·17-s + ⋯ |
Λ(s)=(=(115s/2ΓC(s)L(s)(−0.171+0.985i)Λ(6−s)
Λ(s)=(=(115s/2ΓC(s+5/2)L(s)(−0.171+0.985i)Λ(1−s)
Degree: |
2 |
Conductor: |
115
= 5⋅23
|
Sign: |
−0.171+0.985i
|
Analytic conductor: |
18.4441 |
Root analytic conductor: |
4.29466 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ115(24,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 115, ( :5/2), −0.171+0.985i)
|
Particular Values
L(3) |
≈ |
1.724851075 |
L(21) |
≈ |
1.724851075 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−55.0−9.59i)T |
| 23 | 1+529iT |
good | 2 | 1−8.29iT−32T2 |
| 3 | 1−20.3iT−243T2 |
| 7 | 1−101.iT−1.68e4T2 |
| 11 | 1+362.T+1.61e5T2 |
| 13 | 1+540.iT−3.71e5T2 |
| 17 | 1−561.iT−1.41e6T2 |
| 19 | 1+1.11e3T+2.47e6T2 |
| 29 | 1−7.26e3T+2.05e7T2 |
| 31 | 1−450.T+2.86e7T2 |
| 37 | 1−4.96e3iT−6.93e7T2 |
| 41 | 1−7.55e3T+1.15e8T2 |
| 43 | 1−1.60e4iT−1.47e8T2 |
| 47 | 1+1.94e4iT−2.29e8T2 |
| 53 | 1−5.35e3iT−4.18e8T2 |
| 59 | 1+4.93e4T+7.14e8T2 |
| 61 | 1−1.63e3T+8.44e8T2 |
| 67 | 1+3.44e4iT−1.35e9T2 |
| 71 | 1+5.39e3T+1.80e9T2 |
| 73 | 1−5.24e4iT−2.07e9T2 |
| 79 | 1+4.43e3T+3.07e9T2 |
| 83 | 1−9.18e4iT−3.93e9T2 |
| 89 | 1−8.87e4T+5.58e9T2 |
| 97 | 1−5.56e4iT−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
−13.78967302429692143960492154024, −12.67172748455404224041258252228, −10.81879547056388132248357173632, −10.04942344640460667265339055995, −8.964950815110959858449691136681, −8.074439358485842620303104597548, −6.42108048482741060673798472673, −5.52656829008007783805803883415, −4.72517758308867770575037365780, −2.70189249286172917619152542473,
0.62931305016555509234678932482, 1.71914438263393115023555760907, 2.66246861983355665848731167472, 4.53480990403258999517752986462, 6.32691510820317620921661145660, 7.38218327996330290651196602103, 8.896503689572419184503153731089, 10.08197249101645712285869536207, 10.85161134779635722526557843428, 12.09142181119958421857956983013