L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (0.866 + 1.5i)19-s + (−0.499 − 0.866i)21-s + (−0.707 + 0.707i)27-s + (0.258 − 0.965i)28-s − 36-s + (0.448 − 1.67i)37-s + (−0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (−0.866 + 0.499i)49-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (0.866 + 1.5i)19-s + (−0.499 − 0.866i)21-s + (−0.707 + 0.707i)27-s + (0.258 − 0.965i)28-s − 36-s + (0.448 − 1.67i)37-s + (−0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (−0.866 + 0.499i)49-s + ⋯ |
Λ(s)=(=(525s/2ΓC(s)L(s)(0.710−0.703i)Λ(1−s)
Λ(s)=(=(525s/2ΓC(s)L(s)(0.710−0.703i)Λ(1−s)
Degree: |
2 |
Conductor: |
525
= 3⋅52⋅7
|
Sign: |
0.710−0.703i
|
Analytic conductor: |
0.262009 |
Root analytic conductor: |
0.511868 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ525(332,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 525, ( :0), 0.710−0.703i)
|
Particular Values
L(21) |
≈ |
0.5735010221 |
L(21) |
≈ |
0.5735010221 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.965−0.258i)T |
| 5 | 1 |
| 7 | 1+(−0.258−0.965i)T |
good | 2 | 1+(0.866+0.5i)T2 |
| 11 | 1+(0.5+0.866i)T2 |
| 13 | 1+(−0.707−0.707i)T+iT2 |
| 17 | 1+(0.866−0.5i)T2 |
| 19 | 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 |
| 23 | 1+(−0.866−0.5i)T2 |
| 29 | 1+T2 |
| 31 | 1+(0.5+0.866i)T2 |
| 37 | 1+(−0.448+1.67i)T+(−0.866−0.5i)T2 |
| 41 | 1+T2 |
| 43 | 1−iT2 |
| 47 | 1+(−0.866−0.5i)T2 |
| 53 | 1+(0.866−0.5i)T2 |
| 59 | 1+(0.5+0.866i)T2 |
| 61 | 1+(1.5−0.866i)T+(0.5−0.866i)T2 |
| 67 | 1+(1.67−0.448i)T+(0.866−0.5i)T2 |
| 71 | 1−T2 |
| 73 | 1+(−0.965+0.258i)T+(0.866−0.5i)T2 |
| 79 | 1+(−0.866+0.5i)T+(0.5−0.866i)T2 |
| 83 | 1−iT2 |
| 89 | 1+(0.5−0.866i)T2 |
| 97 | 1+(−0.707+0.707i)T−iT2 |
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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
−11.16193192017319104432118326329, −10.30471748182133627122026676919, −9.438623159824397646965134463012, −8.806169123518105923750640424579, −7.61424063678983368829928099433, −6.11541754742608842168433919976, −5.71444109645330787947357036142, −4.71368681971134051877329674142, −3.72425329598832982644511654373, −1.55080816345337616069209461415,
0.957285847337279441362749789493, 3.28069687518321590505497826757, 4.50189451263717730946963683119, 5.14959005626572939246939861449, 6.41009233705777843818685920659, 7.42048689624225360458138026071, 8.092928672526680151973884199235, 9.277509282498893479671739447032, 10.20537933343363281521056356123, 11.01607537336673876195819766481