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 + (−3.14 − 0.352i)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 + (−0.993 − 0.111i)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)(0.735+0.677i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.735+0.677i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.735+0.677i
|
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), 0.735+0.677i)
|
Particular Values
L(1) |
≈ |
1.24699−0.486391i |
L(21) |
≈ |
1.24699−0.486391i |
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
−9.962243242811442181339363916482, −9.245062985684059501737798612038, −8.156926715578792681103493415972, −7.29072086286097160335133925322, −6.91812755796347676966571454238, −5.89676884818298630821351570699, −4.75503932989340113318611190256, −2.98392017182146781851145926891, −1.96452431896528926121047370542, −1.19101043850434257315231746232,
1.08369852473739588665449664551, 2.60911874337414455949775156647, 3.61510217963040851941385707144, 5.30328053927552948010635409917, 5.64476662776271420731604955683, 6.79910587637447088969232268865, 7.68004466369039999853060123697, 8.762337901640129922121908876432, 9.386289710593094802873034486127, 9.920395147892714654472806808863