L(s) = 1 | + (0.352 + 0.352i)2-s + (0.0655 − 0.158i)3-s − 1.75i·4-s + (2.22 − 0.233i)5-s + (0.0789 − 0.0327i)6-s + (−3.57 + 1.48i)7-s + (1.32 − 1.32i)8-s + (2.10 + 2.10i)9-s + (0.867 + 0.702i)10-s + (−2.71 + 1.12i)11-s + (−0.277 − 0.114i)12-s + 0.983·13-s + (−1.78 − 0.739i)14-s + (0.108 − 0.367i)15-s − 2.56·16-s + (−3.97 − 1.11i)17-s + ⋯ |
L(s) = 1 | + (0.249 + 0.249i)2-s + (0.0378 − 0.0913i)3-s − 0.875i·4-s + (0.994 − 0.104i)5-s + (0.0322 − 0.0133i)6-s + (−1.35 + 0.560i)7-s + (0.468 − 0.468i)8-s + (0.700 + 0.700i)9-s + (0.274 + 0.222i)10-s + (−0.817 + 0.338i)11-s + (−0.0799 − 0.0331i)12-s + 0.272·13-s + (−0.477 − 0.197i)14-s + (0.0281 − 0.0948i)15-s − 0.641·16-s + (−0.962 − 0.269i)17-s + ⋯ |
Λ(s)=(=(85s/2ΓC(s)L(s)(0.992+0.122i)Λ(2−s)
Λ(s)=(=(85s/2ΓC(s+1/2)L(s)(0.992+0.122i)Λ(1−s)
Degree: |
2 |
Conductor: |
85
= 5⋅17
|
Sign: |
0.992+0.122i
|
Analytic conductor: |
0.678728 |
Root analytic conductor: |
0.823849 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ85(9,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 85, ( :1/2), 0.992+0.122i)
|
Particular Values
L(1) |
≈ |
1.12583−0.0689475i |
L(21) |
≈ |
1.12583−0.0689475i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−2.22+0.233i)T |
| 17 | 1+(3.97+1.11i)T |
good | 2 | 1+(−0.352−0.352i)T+2iT2 |
| 3 | 1+(−0.0655+0.158i)T+(−2.12−2.12i)T2 |
| 7 | 1+(3.57−1.48i)T+(4.94−4.94i)T2 |
| 11 | 1+(2.71−1.12i)T+(7.77−7.77i)T2 |
| 13 | 1−0.983T+13T2 |
| 19 | 1+(1.81−1.81i)T−19iT2 |
| 23 | 1+(1.15+2.77i)T+(−16.2+16.2i)T2 |
| 29 | 1+(−0.210+0.507i)T+(−20.5−20.5i)T2 |
| 31 | 1+(−6.98−2.89i)T+(21.9+21.9i)T2 |
| 37 | 1+(3.76−9.09i)T+(−26.1−26.1i)T2 |
| 41 | 1+(2.83+6.84i)T+(−28.9+28.9i)T2 |
| 43 | 1+(−5.85+5.85i)T−43iT2 |
| 47 | 1−10.9T+47T2 |
| 53 | 1+(2.53+2.53i)T+53iT2 |
| 59 | 1+(−0.216−0.216i)T+59iT2 |
| 61 | 1+(2.60+6.28i)T+(−43.1+43.1i)T2 |
| 67 | 1+5.09iT−67T2 |
| 71 | 1+(3.33+1.38i)T+(50.2+50.2i)T2 |
| 73 | 1+(−4.62−1.91i)T+(51.6+51.6i)T2 |
| 79 | 1+(11.4−4.76i)T+(55.8−55.8i)T2 |
| 83 | 1+(5.74+5.74i)T+83iT2 |
| 89 | 1+13.2iT−89T2 |
| 97 | 1+(−4.18−1.73i)T+(68.5+68.5i)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
−13.97833987045927277548935689243, −13.33072018912523887053738152390, −12.52350576580854581932063491295, −10.47096676235807926491150419323, −10.05710824637119186164645388167, −8.859276416804748016895937914233, −6.88672146216983981875664605944, −6.01793249832006547079345655237, −4.80572087540721883376827377287, −2.28553238167153591848681809117,
2.80434187622316532833073440321, 4.17277877480860633602450274244, 6.16517849060474482741770615816, 7.20526374946343897246947009936, 8.875250355760475775455998716885, 9.918971402220433003942939858254, 10.93048220805254405152833006369, 12.54889627371734169776664127169, 13.16734805802509375354958571629, 13.76370139201821994115295430954