L(s) = 1 | + 8i·3-s − 12i·5-s − 32·7-s − 37·9-s + 8i·11-s − 20i·13-s + 96·15-s − 98·17-s − 88i·19-s − 256i·21-s + 32·23-s − 19·25-s − 80i·27-s + 172i·29-s − 256·31-s + ⋯ |
L(s) = 1 | + 1.53i·3-s − 1.07i·5-s − 1.72·7-s − 1.37·9-s + 0.219i·11-s − 0.426i·13-s + 1.65·15-s − 1.39·17-s − 1.06i·19-s − 2.66i·21-s + 0.290·23-s − 0.151·25-s − 0.570i·27-s + 1.10i·29-s − 1.48·31-s + ⋯ |
Λ(s)=(=(128s/2ΓC(s)L(s)(−0.707+0.707i)Λ(4−s)
Λ(s)=(=(128s/2ΓC(s+3/2)L(s)(−0.707+0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
128
= 27
|
Sign: |
−0.707+0.707i
|
Analytic conductor: |
7.55224 |
Root analytic conductor: |
2.74813 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ128(65,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 128, ( :3/2), −0.707+0.707i)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
good | 3 | 1−8iT−27T2 |
| 5 | 1+12iT−125T2 |
| 7 | 1+32T+343T2 |
| 11 | 1−8iT−1.33e3T2 |
| 13 | 1+20iT−2.19e3T2 |
| 17 | 1+98T+4.91e3T2 |
| 19 | 1+88iT−6.85e3T2 |
| 23 | 1−32T+1.21e4T2 |
| 29 | 1−172iT−2.43e4T2 |
| 31 | 1+256T+2.97e4T2 |
| 37 | 1+92iT−5.06e4T2 |
| 41 | 1+102T+6.89e4T2 |
| 43 | 1−296iT−7.95e4T2 |
| 47 | 1+320T+1.03e5T2 |
| 53 | 1+76iT−1.48e5T2 |
| 59 | 1+408iT−2.05e5T2 |
| 61 | 1−636iT−2.26e5T2 |
| 67 | 1−552iT−3.00e5T2 |
| 71 | 1+416T+3.57e5T2 |
| 73 | 1+138T+3.89e5T2 |
| 79 | 1+64T+4.93e5T2 |
| 83 | 1−392iT−5.71e5T2 |
| 89 | 1−582T+7.04e5T2 |
| 97 | 1−238T+9.12e5T2 |
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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
−12.78634401478974716182020570086, −11.20949164116986533597062722664, −10.21450542201675788116760123957, −9.212338049798465062086091506886, −8.903006379956564090130312162996, −6.82132329402451852400860869549, −5.36702368536121864314502922349, −4.35786704626479886346482269382, −3.12876406372654968398368489262, 0,
2.20797048657158391561318135753, 3.46338842835132231093544429303, 6.15864662847546624487554837769, 6.63439182096268572137556698852, 7.48396867514535230660036209520, 8.953014507280457733055146880209, 10.20760603513653161292808924193, 11.37923953594496628618316219292, 12.45092757056855816104567684050