L(s) = 1 | + (1.38 + 0.295i)2-s − 1.66i·3-s + (1.82 + 0.817i)4-s + (2.22 + 0.220i)5-s + (0.491 − 2.29i)6-s + (2.28 + 1.67i)8-s + 0.236·9-s + (3.01 + 0.962i)10-s + 4.81i·11-s + (1.35 − 3.03i)12-s − 2.14·13-s + (0.366 − 3.69i)15-s + (2.66 + 2.98i)16-s + 5.02·17-s + (0.326 + 0.0698i)18-s − 1.36·19-s + ⋯ |
L(s) = 1 | + (0.977 + 0.209i)2-s − 0.959i·3-s + (0.912 + 0.408i)4-s + (0.995 + 0.0986i)5-s + (0.200 − 0.938i)6-s + (0.807 + 0.590i)8-s + 0.0787·9-s + (0.952 + 0.304i)10-s + 1.45i·11-s + (0.392 − 0.875i)12-s − 0.594·13-s + (0.0946 − 0.955i)15-s + (0.665 + 0.746i)16-s + 1.21·17-s + (0.0770 + 0.0164i)18-s − 0.312·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.995+0.0982i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.995+0.0982i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.995+0.0982i
|
Analytic conductor: |
7.82533 |
Root analytic conductor: |
2.79738 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ980(979,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 980, ( :1/2), 0.995+0.0982i)
|
Particular Values
L(1) |
≈ |
3.58802−0.176689i |
L(21) |
≈ |
3.58802−0.176689i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.38−0.295i)T |
| 5 | 1+(−2.22−0.220i)T |
| 7 | 1 |
good | 3 | 1+1.66iT−3T2 |
| 11 | 1−4.81iT−11T2 |
| 13 | 1+2.14T+13T2 |
| 17 | 1−5.02T+17T2 |
| 19 | 1+1.36T+19T2 |
| 23 | 1+5.18T+23T2 |
| 29 | 1+6.43T+29T2 |
| 31 | 1+4.62T+31T2 |
| 37 | 1+9.82iT−37T2 |
| 41 | 1+4.71iT−41T2 |
| 43 | 1+0.141T+43T2 |
| 47 | 1+2.55iT−47T2 |
| 53 | 1−4.84iT−53T2 |
| 59 | 1+14.1T+59T2 |
| 61 | 1+10.1iT−61T2 |
| 67 | 1−9.64T+67T2 |
| 71 | 1+9.58iT−71T2 |
| 73 | 1+1.67T+73T2 |
| 79 | 1+11.8iT−79T2 |
| 83 | 1+0.811iT−83T2 |
| 89 | 1−16.0iT−89T2 |
| 97 | 1−1.76T+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
−10.05532637526589518617796325310, −9.285637756814458982618912966622, −7.65160036644781704879301493188, −7.46653740517553948206642635565, −6.51641195975747280694047204974, −5.74200232584618368704561681617, −4.89871603092514844148512813012, −3.74543591275488929093842542381, −2.22032711936415080555945160840, −1.79718388630141139602623651953,
1.51668331080445335445414894223, 2.91765180050905360426082725672, 3.73022051128516001545653514066, 4.78334224037148177462912763743, 5.59577024403653414120460604442, 6.12020209320716014670980442669, 7.30887767122884159584079868589, 8.450628062825298781845394366710, 9.670184863883879296452123384477, 9.976519102351300494964615980161