L(s) = 1 | + (1.37 − 0.792i)2-s + (−0.0510 + 0.190i)3-s + (0.255 − 0.442i)4-s + (2.23 + 0.0672i)5-s + (0.0809 + 0.302i)6-s + (−0.274 + 0.474i)7-s + 2.35i·8-s + (2.56 + 1.48i)9-s + (3.12 − 1.67i)10-s + (0.0396 − 0.147i)11-s + (0.0713 + 0.0713i)12-s + 0.868i·14-s + (−0.126 + 0.422i)15-s + (2.38 + 4.12i)16-s + (−3.03 + 0.813i)17-s + 4.69·18-s + ⋯ |
L(s) = 1 | + (0.970 − 0.560i)2-s + (−0.0294 + 0.110i)3-s + (0.127 − 0.221i)4-s + (0.999 + 0.0300i)5-s + (0.0330 + 0.123i)6-s + (−0.103 + 0.179i)7-s + 0.834i·8-s + (0.854 + 0.493i)9-s + (0.986 − 0.530i)10-s + (0.0119 − 0.0446i)11-s + (0.0205 + 0.0205i)12-s + 0.232i·14-s + (−0.0327 + 0.109i)15-s + (0.595 + 1.03i)16-s + (−0.736 + 0.197i)17-s + 1.10·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(0.987−0.156i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(0.987−0.156i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
0.987−0.156i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(418,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), 0.987−0.156i)
|
Particular Values
L(1) |
≈ |
2.94897+0.231545i |
L(21) |
≈ |
2.94897+0.231545i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−2.23−0.0672i)T |
| 13 | 1 |
good | 2 | 1+(−1.37+0.792i)T+(1−1.73i)T2 |
| 3 | 1+(0.0510−0.190i)T+(−2.59−1.5i)T2 |
| 7 | 1+(0.274−0.474i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−0.0396+0.147i)T+(−9.52−5.5i)T2 |
| 17 | 1+(3.03−0.813i)T+(14.7−8.5i)T2 |
| 19 | 1+(4.40−1.18i)T+(16.4−9.5i)T2 |
| 23 | 1+(−3.41−0.916i)T+(19.9+11.5i)T2 |
| 29 | 1+(2.02−1.17i)T+(14.5−25.1i)T2 |
| 31 | 1+(−6.60+6.60i)T−31iT2 |
| 37 | 1+(3.40+5.89i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−3.45−0.926i)T+(35.5+20.5i)T2 |
| 43 | 1+(1.84+6.86i)T+(−37.2+21.5i)T2 |
| 47 | 1−9.13T+47T2 |
| 53 | 1+(3.70+3.70i)T+53iT2 |
| 59 | 1+(0.985+3.67i)T+(−51.0+29.5i)T2 |
| 61 | 1+(3.92−6.79i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−4.23+2.44i)T+(33.5−58.0i)T2 |
| 71 | 1+(4.04+15.1i)T+(−61.4+35.5i)T2 |
| 73 | 1+3.91iT−73T2 |
| 79 | 1+11.1iT−79T2 |
| 83 | 1+13.4T+83T2 |
| 89 | 1+(−8.78−2.35i)T+(77.0+44.5i)T2 |
| 97 | 1+(−6.55−3.78i)T+(48.5+84.0i)T2 |
show more | |
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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
−10.61506731167254663662601980635, −9.444178939021150104166061659990, −8.714875971083100195737748393279, −7.59582039962938718315204868751, −6.45587313036957912057017153516, −5.62234427335603430456528500327, −4.71294917191507755181049573882, −3.95849822837942982539647982324, −2.61122444756349376472768001558, −1.83253976670902487318384438658,
1.22455510612709162799641902281, 2.77498609390830597851609265557, 4.15464942146242399447594102623, 4.82062764923303978087094858902, 5.82346078002596773261416076253, 6.73623527416839077933245510500, 6.94872867994310899147990966016, 8.528491271192588476546170971656, 9.437406426790504805879771125767, 10.09997800301411522778980792414