L(s) = 1 | + (−0.800 − 1.16i)2-s + 2.62i·3-s + (−0.719 + 1.86i)4-s + (3.06 − 2.10i)6-s − 0.269·7-s + (2.75 − 0.654i)8-s − 3.89·9-s − 4.58i·11-s + (−4.90 − 1.88i)12-s − 1.55i·13-s + (0.215 + 0.313i)14-s + (−2.96 − 2.68i)16-s − 0.609·17-s + (3.12 + 4.54i)18-s − 6.69i·19-s + ⋯ |
L(s) = 1 | + (−0.565 − 0.824i)2-s + 1.51i·3-s + (−0.359 + 0.933i)4-s + (1.25 − 0.858i)6-s − 0.101·7-s + (0.972 − 0.231i)8-s − 1.29·9-s − 1.38i·11-s + (−1.41 − 0.545i)12-s − 0.430i·13-s + (0.0575 + 0.0839i)14-s + (−0.741 − 0.671i)16-s − 0.147·17-s + (0.735 + 1.07i)18-s − 1.53i·19-s + ⋯ |
Λ(s)=(=(1000s/2ΓC(s)L(s)(0.231+0.972i)Λ(2−s)
Λ(s)=(=(1000s/2ΓC(s+1/2)L(s)(0.231+0.972i)Λ(1−s)
Degree: |
2 |
Conductor: |
1000
= 23⋅53
|
Sign: |
0.231+0.972i
|
Analytic conductor: |
7.98504 |
Root analytic conductor: |
2.82578 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1000(501,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1000, ( :1/2), 0.231+0.972i)
|
Particular Values
L(1) |
≈ |
0.597556−0.472123i |
L(21) |
≈ |
0.597556−0.472123i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.800+1.16i)T |
| 5 | 1 |
good | 3 | 1−2.62iT−3T2 |
| 7 | 1+0.269T+7T2 |
| 11 | 1+4.58iT−11T2 |
| 13 | 1+1.55iT−13T2 |
| 17 | 1+0.609T+17T2 |
| 19 | 1+6.69iT−19T2 |
| 23 | 1+3.52T+23T2 |
| 29 | 1+7.73iT−29T2 |
| 31 | 1+3.34T+31T2 |
| 37 | 1+5.32iT−37T2 |
| 41 | 1−4.38T+41T2 |
| 43 | 1−12.2iT−43T2 |
| 47 | 1−9.54T+47T2 |
| 53 | 1+10.6iT−53T2 |
| 59 | 1−10.2iT−59T2 |
| 61 | 1+13.2iT−61T2 |
| 67 | 1+3.93iT−67T2 |
| 71 | 1−6.95T+71T2 |
| 73 | 1+5.93T+73T2 |
| 79 | 1−10.1T+79T2 |
| 83 | 1+6.52iT−83T2 |
| 89 | 1−6.69T+89T2 |
| 97 | 1−3.99T+97T2 |
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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
−9.713221490170034665272327132948, −9.282034029983826003205618189377, −8.502921191288745479663416344050, −7.69556145356589666874966450644, −6.23789004048709129090366807212, −5.14417773083131138162404264736, −4.21893680979658266133672881122, −3.44031351647366409487204547096, −2.55132238845248669318751086039, −0.44203805577804098941225018369,
1.38461769662668654586874447687, 2.14636449955759814691807966042, 4.09062616720741030860980087805, 5.35172069930406926219673254331, 6.21604334041925689043479902887, 7.00281388768813495319184234983, 7.46856272630132693554146740940, 8.222867691749538731442877689510, 9.100074632878273339320733231916, 9.988506618297499728636860021126