L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.00i·9-s + (−1.41 + 1.41i)19-s + i·25-s + (0.707 + 0.707i)27-s + (1.41 + 1.41i)43-s + 49-s − 2.00i·57-s + (−1.41 + 1.41i)67-s + 2i·73-s + (−0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.00i·9-s + (−1.41 + 1.41i)19-s + i·25-s + (0.707 + 0.707i)27-s + (1.41 + 1.41i)43-s + 49-s − 2.00i·57-s + (−1.41 + 1.41i)67-s + 2i·73-s + (−0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯ |
Λ(s)=(=(3072s/2ΓC(s)L(s)(−0.382−0.923i)Λ(1−s)
Λ(s)=(=(3072s/2ΓC(s)L(s)(−0.382−0.923i)Λ(1−s)
Degree: |
2 |
Conductor: |
3072
= 210⋅3
|
Sign: |
−0.382−0.923i
|
Analytic conductor: |
1.53312 |
Root analytic conductor: |
1.23819 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3072(1793,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3072, ( :0), −0.382−0.923i)
|
Particular Values
L(21) |
≈ |
0.7304416220 |
L(21) |
≈ |
0.7304416220 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(0.707−0.707i)T |
good | 5 | 1−iT2 |
| 7 | 1−T2 |
| 11 | 1−iT2 |
| 13 | 1−iT2 |
| 17 | 1−T2 |
| 19 | 1+(1.41−1.41i)T−iT2 |
| 23 | 1+T2 |
| 29 | 1+iT2 |
| 31 | 1+T2 |
| 37 | 1+iT2 |
| 41 | 1+T2 |
| 43 | 1+(−1.41−1.41i)T+iT2 |
| 47 | 1−T2 |
| 53 | 1−iT2 |
| 59 | 1−iT2 |
| 61 | 1−iT2 |
| 67 | 1+(1.41−1.41i)T−iT2 |
| 71 | 1+T2 |
| 73 | 1−2iT−T2 |
| 79 | 1+T2 |
| 83 | 1+iT2 |
| 89 | 1+T2 |
| 97 | 1+2T+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
−9.188777729223045129511174371523, −8.501288829706252621221386772923, −7.61252048033858873517136605199, −6.71287564274634971063025448154, −5.94094145167146399279470691750, −5.43063973870499996027286541794, −4.32620683929509160746485781991, −3.89702041163973258740634789180, −2.74198755636911149032372621669, −1.36725538656361626900124952109,
0.51063105119599922290331032342, 1.95712338329204378325968463816, 2.72425410088202730598982268643, 4.16230907033899008918432432400, 4.81478377608992146015288375312, 5.75452046609404865511276035549, 6.42785414255528850372055491136, 7.05304989496546062757050854351, 7.79040635287870126992935266952, 8.624354187827936717032718184389