L(s) = 1 | − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s − i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯ |
L(s) = 1 | − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s − i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯ |
Λ(s)=(=(435s/2ΓC(s)L(s)(−0.707+0.707i)Λ(1−s)
Λ(s)=(=(435s/2ΓC(s)L(s)(−0.707+0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
435
= 3⋅5⋅29
|
Sign: |
−0.707+0.707i
|
Analytic conductor: |
0.217093 |
Root analytic conductor: |
0.465932 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ435(434,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 435, ( :0), −0.707+0.707i)
|
Particular Values
L(21) |
≈ |
0.6549346783 |
L(21) |
≈ |
0.6549346783 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.707−0.707i)T |
| 5 | 1+iT |
| 29 | 1−iT |
good | 2 | 1+1.41iT−T2 |
| 7 | 1−T2 |
| 11 | 1+T2 |
| 13 | 1−T2 |
| 17 | 1+1.41iT−T2 |
| 19 | 1−T2 |
| 23 | 1+T2 |
| 31 | 1−T2 |
| 37 | 1−1.41T+T2 |
| 41 | 1+T2 |
| 43 | 1−1.41T+T2 |
| 47 | 1−1.41iT−T2 |
| 53 | 1+T2 |
| 59 | 1−2iT−T2 |
| 61 | 1−T2 |
| 67 | 1−T2 |
| 71 | 1+2iT−T2 |
| 73 | 1+1.41T+T2 |
| 79 | 1−T2 |
| 83 | 1+T2 |
| 89 | 1+T2 |
| 97 | 1+1.41T+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
−11.10925556937784685257595298515, −10.30107898932773866462678532612, −9.355172065686950677158247970746, −9.016485434287188897409415824302, −7.37611956487662343087107882024, −5.93876239627619688569353930404, −4.83169075143660255869303250892, −4.14429183287958096912215365830, −2.82987199154408230938686832926, −1.02405564282509788393045643942,
2.29200787907349334667327287820, 4.20323200289694540433527767246, 5.64776528050153626290529813582, 6.16702982690644972467450122874, 6.97099743013341101755526103891, 7.71690287397927697091285302851, 8.458656965702244809334768731076, 9.932589286928448295718445679514, 10.91973120262719832743795947517, 11.57612340855838048530728519379