L(s) = 1 | + (−1.23 − 0.691i)2-s − 2.08i·3-s + (1.04 + 1.70i)4-s + (1.52 + 1.63i)5-s + (−1.43 + 2.56i)6-s + (−0.107 − 2.82i)8-s − 1.32·9-s + (−0.746 − 3.07i)10-s − 0.775i·11-s + (3.54 − 2.17i)12-s + 4.18·13-s + (3.40 − 3.16i)15-s + (−1.82 + 3.56i)16-s − 4.18·17-s + (1.63 + 0.917i)18-s − 4.88·19-s + ⋯ |
L(s) = 1 | + (−0.872 − 0.488i)2-s − 1.20i·3-s + (0.521 + 0.853i)4-s + (0.680 + 0.732i)5-s + (−0.587 + 1.04i)6-s + (−0.0381 − 0.999i)8-s − 0.442·9-s + (−0.235 − 0.971i)10-s − 0.233i·11-s + (1.02 − 0.626i)12-s + 1.16·13-s + (0.879 − 0.817i)15-s + (−0.455 + 0.890i)16-s − 1.01·17-s + (0.385 + 0.216i)18-s − 1.12·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.148+0.988i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.148+0.988i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.148+0.988i
|
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), 0.148+0.988i)
|
Particular Values
L(1) |
≈ |
0.952902−0.820739i |
L(21) |
≈ |
0.952902−0.820739i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.23+0.691i)T |
| 5 | 1+(−1.52−1.63i)T |
| 7 | 1 |
good | 3 | 1+2.08iT−3T2 |
| 11 | 1+0.775iT−11T2 |
| 13 | 1−4.18T+13T2 |
| 17 | 1+4.18T+17T2 |
| 19 | 1+4.88T+19T2 |
| 23 | 1−1.89T+23T2 |
| 29 | 1−9.98T+29T2 |
| 31 | 1−10.3T+31T2 |
| 37 | 1+3.41iT−37T2 |
| 41 | 1+3.02iT−41T2 |
| 43 | 1−9.19T+43T2 |
| 47 | 1+8.27iT−47T2 |
| 53 | 1−2.59iT−53T2 |
| 59 | 1+4.60T+59T2 |
| 61 | 1+3.26iT−61T2 |
| 67 | 1+2.27T+67T2 |
| 71 | 1−4.41iT−71T2 |
| 73 | 1−2.74T+73T2 |
| 79 | 1−14.2iT−79T2 |
| 83 | 1+2.36iT−83T2 |
| 89 | 1+14.3iT−89T2 |
| 97 | 1+14.4T+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
−9.934679776420637321547321733552, −8.749229077213744150397041341101, −8.349794606413398973609359431360, −7.23923048100569660871740947942, −6.52535724788495087073899980092, −6.15319057573086786674734275376, −4.23098426455564776626970141616, −2.84656628330016052107043151000, −2.09357944728399799488079597539, −0.954542935649508740206825872310,
1.18614043410395345958499512997, 2.63590937095830209847617359923, 4.39338511228907902457448009822, 4.82396994851791454761700344242, 6.11591644991270745190804102738, 6.54952471222072196025010994871, 8.088595862310847988761467420176, 8.757112890681949819193592980131, 9.237355056197465632718698048960, 10.14590844416502734573889860436