L(s) = 1 | + (1.17 + 0.784i)2-s + 2.99·3-s + (0.767 + 1.84i)4-s + i·5-s + (3.52 + 2.35i)6-s + (−0.546 + 2.77i)8-s + 5.98·9-s + (−0.784 + 1.17i)10-s − 2.23i·11-s + (2.30 + 5.53i)12-s − 3.17i·13-s + 2.99i·15-s + (−2.82 + 2.83i)16-s − 3.44i·17-s + (7.04 + 4.70i)18-s − 2.05·19-s + ⋯ |
L(s) = 1 | + (0.831 + 0.555i)2-s + 1.73·3-s + (0.383 + 0.923i)4-s + 0.447i·5-s + (1.43 + 0.960i)6-s + (−0.193 + 0.981i)8-s + 1.99·9-s + (−0.248 + 0.372i)10-s − 0.674i·11-s + (0.664 + 1.59i)12-s − 0.879i·13-s + 0.774i·15-s + (−0.705 + 0.708i)16-s − 0.835i·17-s + (1.66 + 1.10i)18-s − 0.470·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.446−0.894i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.446−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.446−0.894i
|
Analytic conductor: |
7.82533 |
Root analytic conductor: |
2.79738 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ980(391,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 980, ( :1/2), 0.446−0.894i)
|
Particular Values
L(1) |
≈ |
3.75951+2.32501i |
L(21) |
≈ |
3.75951+2.32501i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.17−0.784i)T |
| 5 | 1−iT |
| 7 | 1 |
good | 3 | 1−2.99T+3T2 |
| 11 | 1+2.23iT−11T2 |
| 13 | 1+3.17iT−13T2 |
| 17 | 1+3.44iT−17T2 |
| 19 | 1+2.05T+19T2 |
| 23 | 1−2.66iT−23T2 |
| 29 | 1+7.38T+29T2 |
| 31 | 1−4.89T+31T2 |
| 37 | 1+11.1T+37T2 |
| 41 | 1+1.46iT−41T2 |
| 43 | 1−9.95iT−43T2 |
| 47 | 1+6.12T+47T2 |
| 53 | 1−4.65T+53T2 |
| 59 | 1−7.11T+59T2 |
| 61 | 1−2.53iT−61T2 |
| 67 | 1+0.0527iT−67T2 |
| 71 | 1−0.212iT−71T2 |
| 73 | 1+14.8iT−73T2 |
| 79 | 1+0.461iT−79T2 |
| 83 | 1−10.9T+83T2 |
| 89 | 1+7.02iT−89T2 |
| 97 | 1−0.185iT−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
−9.974221190791594698385576227227, −9.014614757677361745339236335175, −8.314438975265574071866214795018, −7.64269914486441582319086563494, −6.97693236464052709904971093416, −5.87328537060739947108711748792, −4.75520015346138953698494004948, −3.52728377581674454598404043356, −3.15231887217468327294781574358, −2.10229250721194179269481334841,
1.71261683136366183297568567854, 2.30459185909738030456326848123, 3.62848167897110850129297304236, 4.14051130784717188158348762433, 5.14674801621780780245756109250, 6.54598300901966492662224188829, 7.30798714723281555662939104967, 8.435262765175801982664951315154, 9.019375259686658539043780201646, 9.853266004484854632008113056555