L(s) = 1 | + (−0.457 + 2.79i)2-s + 3i·3-s + (−7.58 − 2.55i)4-s + 4.56·5-s + (−8.37 − 1.37i)6-s + 23.0i·7-s + (10.5 − 19.9i)8-s − 9·9-s + (−2.08 + 12.7i)10-s − 9.02·11-s + (7.65 − 22.7i)12-s + (3.63 + 46.7i)13-s + (−64.1 − 10.5i)14-s + 13.7i·15-s + (50.9 + 38.7i)16-s − 59.4·17-s + ⋯ |
L(s) = 1 | + (−0.161 + 0.986i)2-s + 0.577i·3-s + (−0.947 − 0.319i)4-s + 0.408·5-s + (−0.569 − 0.0933i)6-s + 1.24i·7-s + (0.468 − 0.883i)8-s − 0.333·9-s + (−0.0660 + 0.403i)10-s − 0.247·11-s + (0.184 − 0.547i)12-s + (0.0776 + 0.996i)13-s + (−1.22 − 0.200i)14-s + 0.235i·15-s + (0.796 + 0.604i)16-s − 0.847·17-s + ⋯ |
Λ(s)=(=(312s/2ΓC(s)L(s)(−0.398+0.917i)Λ(4−s)
Λ(s)=(=(312s/2ΓC(s+3/2)L(s)(−0.398+0.917i)Λ(1−s)
Degree: |
2 |
Conductor: |
312
= 23⋅3⋅13
|
Sign: |
−0.398+0.917i
|
Analytic conductor: |
18.4085 |
Root analytic conductor: |
4.29052 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ312(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 312, ( :3/2), −0.398+0.917i)
|
Particular Values
L(2) |
≈ |
0.7130169423 |
L(21) |
≈ |
0.7130169423 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.457−2.79i)T |
| 3 | 1−3iT |
| 13 | 1+(−3.63−46.7i)T |
good | 5 | 1−4.56T+125T2 |
| 7 | 1−23.0iT−343T2 |
| 11 | 1+9.02T+1.33e3T2 |
| 17 | 1+59.4T+4.91e3T2 |
| 19 | 1+27.0T+6.85e3T2 |
| 23 | 1−35.7T+1.21e4T2 |
| 29 | 1−142.iT−2.43e4T2 |
| 31 | 1+303.iT−2.97e4T2 |
| 37 | 1+241.T+5.06e4T2 |
| 41 | 1+280.iT−6.89e4T2 |
| 43 | 1−198.iT−7.95e4T2 |
| 47 | 1+319.iT−1.03e5T2 |
| 53 | 1−531.iT−1.48e5T2 |
| 59 | 1−378.T+2.05e5T2 |
| 61 | 1+878.iT−2.26e5T2 |
| 67 | 1−247.T+3.00e5T2 |
| 71 | 1+182.iT−3.57e5T2 |
| 73 | 1−815.iT−3.89e5T2 |
| 79 | 1+817.T+4.93e5T2 |
| 83 | 1+246.T+5.71e5T2 |
| 89 | 1−355.iT−7.04e5T2 |
| 97 | 1−506.iT−9.12e5T2 |
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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
−11.86341128190643033931918963455, −10.77643384248045134944369072696, −9.595513846489019043627249355772, −9.066658977127065715075133798932, −8.286208200466808726567463284906, −6.90302017674434340206138071117, −5.93937850517197410566779460718, −5.14459537211193363452742860209, −3.99379012740446711723007237151, −2.16510616306739711483879608532,
0.27457285293938218534573578656, 1.54344962316151436482886648253, 2.91897548479408206122947278407, 4.16792742517924334939511050684, 5.41409990397762997226500909972, 6.83903987545360642265736894603, 7.87305382654675183100319402881, 8.747109150715055653063470274764, 10.01814345507838914477183589048, 10.55925485277392536937518923627