L(s) = 1 | + (1.73 + i)3-s + (−1 − 1.73i)4-s + 4i·7-s + (0.499 + 0.866i)9-s + 3·11-s − 3.99i·12-s + (1.73 − i)13-s + (−1.99 + 3.46i)16-s + (5.19 + 3i)17-s + (3.5 + 2.59i)19-s + (−4 + 6.92i)21-s − 4.00i·27-s + (6.92 − 4i)28-s + (−1.5 − 2.59i)29-s − 7·31-s + ⋯ |
L(s) = 1 | + (0.999 + 0.577i)3-s + (−0.5 − 0.866i)4-s + 1.51i·7-s + (0.166 + 0.288i)9-s + 0.904·11-s − 1.15i·12-s + (0.480 − 0.277i)13-s + (−0.499 + 0.866i)16-s + (1.26 + 0.727i)17-s + (0.802 + 0.596i)19-s + (−0.872 + 1.51i)21-s − 0.769i·27-s + (1.30 − 0.755i)28-s + (−0.278 − 0.482i)29-s − 1.25·31-s + ⋯ |
Λ(s)=(=(475s/2ΓC(s)L(s)(0.846−0.532i)Λ(2−s)
Λ(s)=(=(475s/2ΓC(s+1/2)L(s)(0.846−0.532i)Λ(1−s)
Degree: |
2 |
Conductor: |
475
= 52⋅19
|
Sign: |
0.846−0.532i
|
Analytic conductor: |
3.79289 |
Root analytic conductor: |
1.94753 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ475(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 475, ( :1/2), 0.846−0.532i)
|
Particular Values
L(1) |
≈ |
1.76655+0.509454i |
L(21) |
≈ |
1.76655+0.509454i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 19 | 1+(−3.5−2.59i)T |
good | 2 | 1+(1+1.73i)T2 |
| 3 | 1+(−1.73−i)T+(1.5+2.59i)T2 |
| 7 | 1−4iT−7T2 |
| 11 | 1−3T+11T2 |
| 13 | 1+(−1.73+i)T+(6.5−11.2i)T2 |
| 17 | 1+(−5.19−3i)T+(8.5+14.7i)T2 |
| 23 | 1+(11.5−19.9i)T2 |
| 29 | 1+(1.5+2.59i)T+(−14.5+25.1i)T2 |
| 31 | 1+7T+31T2 |
| 37 | 1+8iT−37T2 |
| 41 | 1+(−3+5.19i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−3.46−2i)T+(21.5+37.2i)T2 |
| 47 | 1+(5.19−3i)T+(23.5−40.7i)T2 |
| 53 | 1+(5.19−3i)T+(26.5−45.8i)T2 |
| 59 | 1+(7.5−12.9i)T+(−29.5−51.0i)T2 |
| 61 | 1+(2.5+4.33i)T+(−30.5+52.8i)T2 |
| 67 | 1+(1.73−i)T+(33.5−58.0i)T2 |
| 71 | 1+(−1.5+2.59i)T+(−35.5−61.4i)T2 |
| 73 | 1+(6.92+4i)T+(36.5+63.2i)T2 |
| 79 | 1+(−2.5+4.33i)T+(−39.5−68.4i)T2 |
| 83 | 1−12iT−83T2 |
| 89 | 1+(7.5+12.9i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−6.92−4i)T+(48.5+84.0i)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
−10.91381483453929153099001190199, −9.844501474706357922007565514408, −9.228234855902200390235690546355, −8.783436304268985795564881361405, −7.77464208771795972208962237532, −5.96004480878040017705807850892, −5.65087247543136885852418416393, −4.14581044886372423467508100544, −3.18598208237196360298998038068, −1.68963123435614751517504654443,
1.26052134014176786655065684227, 3.14190017737766234071291479024, 3.73746216017072337283333209722, 4.94418267041729288861672119069, 6.78498553458026951830260153804, 7.47027233913951022615622415120, 8.030173661203425597557555437492, 9.086983046505120545913033479251, 9.704515666828232234332986424584, 11.04334793401683353504668690531