L(s) = 1 | + 1.41·2-s + (0.707 + 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (0.707 + 0.707i)12-s − 0.999·16-s + 1.41·17-s + 1.41i·18-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·32-s + 2.00·34-s + 1.00i·36-s + 1.41i·37-s + ⋯ |
L(s) = 1 | + 1.41·2-s + (0.707 + 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (0.707 + 0.707i)12-s − 0.999·16-s + 1.41·17-s + 1.41i·18-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·32-s + 2.00·34-s + 1.00i·36-s + 1.41i·37-s + ⋯ |
Λ(s)=(=(2175s/2ΓC(s)L(s)(0.707−0.707i)Λ(1−s)
Λ(s)=(=(2175s/2ΓC(s)L(s)(0.707−0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
2175
= 3⋅52⋅29
|
Sign: |
0.707−0.707i
|
Analytic conductor: |
1.08546 |
Root analytic conductor: |
1.04185 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2175(1826,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2175, ( :0), 0.707−0.707i)
|
Particular Values
L(21) |
≈ |
2.935000948 |
L(21) |
≈ |
2.935000948 |
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 |
| 29 | 1+iT |
good | 2 | 1−1.41T+T2 |
| 7 | 1+T2 |
| 11 | 1+T2 |
| 13 | 1+T2 |
| 17 | 1−1.41T+T2 |
| 19 | 1−T2 |
| 23 | 1−T2 |
| 31 | 1−T2 |
| 37 | 1−1.41iT−T2 |
| 41 | 1+T2 |
| 43 | 1+1.41iT−T2 |
| 47 | 1+1.41T+T2 |
| 53 | 1−T2 |
| 59 | 1+2iT−T2 |
| 61 | 1−T2 |
| 67 | 1+T2 |
| 71 | 1+2iT−T2 |
| 73 | 1−1.41iT−T2 |
| 79 | 1−T2 |
| 83 | 1−T2 |
| 89 | 1+T2 |
| 97 | 1+1.41iT−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
−9.556974442679638383688155829194, −8.463113385973294927159560564195, −7.87807054493720087823055555097, −6.82581002725355237866839870875, −5.90567529884247538508891719902, −5.12524807159713461958350129817, −4.53279872115235031684175581565, −3.54128174111645860228937893763, −3.12826316079474454733365428903, −1.97754050937281310951466859758,
1.47067445861492531573189272021, 2.72157005976622050768543625316, 3.34302047631195795110182512882, 4.15364416745678358955646277686, 5.19891902844400130386375685940, 5.91704177012419885978283356177, 6.69436151736711633518903983827, 7.45840512875108568941140390961, 8.210828407104074380499983712850, 9.109776717196761230132872383499