L(s) = 1 | + (−0.885 − 0.511i)2-s + (−0.721 − 2.69i)3-s + (−0.477 − 0.826i)4-s + (1.45 + 1.69i)5-s + (−0.737 + 2.75i)6-s + (0.481 + 0.834i)7-s + 3.02i·8-s + (−4.12 + 2.38i)9-s + (−0.423 − 2.24i)10-s + (0.430 + 1.60i)11-s + (−1.88 + 1.88i)12-s − 0.985i·14-s + (3.51 − 5.14i)15-s + (0.590 − 1.02i)16-s + (7.00 + 1.87i)17-s + 4.87·18-s + ⋯ |
L(s) = 1 | + (−0.626 − 0.361i)2-s + (−0.416 − 1.55i)3-s + (−0.238 − 0.413i)4-s + (0.651 + 0.758i)5-s + (−0.301 + 1.12i)6-s + (0.182 + 0.315i)7-s + 1.06i·8-s + (−1.37 + 0.794i)9-s + (−0.133 − 0.710i)10-s + (0.129 + 0.484i)11-s + (−0.542 + 0.542i)12-s − 0.263i·14-s + (0.907 − 1.32i)15-s + (0.147 − 0.255i)16-s + (1.69 + 0.455i)17-s + 1.14·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(0.499+0.866i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(0.499+0.866i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
0.499+0.866i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(657,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), 0.499+0.866i)
|
Particular Values
L(1) |
≈ |
0.885770−0.511672i |
L(21) |
≈ |
0.885770−0.511672i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−1.45−1.69i)T |
| 13 | 1 |
good | 2 | 1+(0.885+0.511i)T+(1+1.73i)T2 |
| 3 | 1+(0.721+2.69i)T+(−2.59+1.5i)T2 |
| 7 | 1+(−0.481−0.834i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−0.430−1.60i)T+(−9.52+5.5i)T2 |
| 17 | 1+(−7.00−1.87i)T+(14.7+8.5i)T2 |
| 19 | 1+(−2.64−0.707i)T+(16.4+9.5i)T2 |
| 23 | 1+(−3.72+0.997i)T+(19.9−11.5i)T2 |
| 29 | 1+(−0.253−0.146i)T+(14.5+25.1i)T2 |
| 31 | 1+(−0.125−0.125i)T+31iT2 |
| 37 | 1+(2.04−3.53i)T+(−18.5−32.0i)T2 |
| 41 | 1+(6.69−1.79i)T+(35.5−20.5i)T2 |
| 43 | 1+(2.05−7.67i)T+(−37.2−21.5i)T2 |
| 47 | 1−7.84T+47T2 |
| 53 | 1+(1.99−1.99i)T−53iT2 |
| 59 | 1+(1.30−4.87i)T+(−51.0−29.5i)T2 |
| 61 | 1+(1.04+1.80i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−6.32−3.64i)T+(33.5+58.0i)T2 |
| 71 | 1+(−3.37+12.6i)T+(−61.4−35.5i)T2 |
| 73 | 1−3.22iT−73T2 |
| 79 | 1+13.5iT−79T2 |
| 83 | 1−8.56T+83T2 |
| 89 | 1+(0.500−0.134i)T+(77.0−44.5i)T2 |
| 97 | 1+(6.50−3.75i)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
−10.10540690398641762241766359721, −9.336305654792670622075260532991, −8.239074990246099924466561499933, −7.52397775851728811484150382439, −6.63269200968377286678691061581, −5.81710869885207503797804591678, −5.15505074447522236999876374774, −3.02897189988179498819880876100, −1.89512714422108636181565133897, −1.14389057678432726791314157561,
0.840239350458734116899931495616, 3.23980569404971541847241050644, 4.05201414963801530814618413955, 5.12841354302598861045524134927, 5.61369022554950875773990913031, 7.00509114769537421685174642374, 8.072509064264542113481938577206, 8.904527544627447795053493277293, 9.503655844408591204577677543410, 10.02631289238906676012390090906