L(s) = 1 | + (−0.255 − 1.39i)2-s + 3.18i·3-s + (−1.86 + 0.711i)4-s + (1.80 + 1.31i)5-s + (4.43 − 0.815i)6-s + (1.46 + 2.41i)8-s − 7.15·9-s + (1.36 − 2.85i)10-s + 4.51i·11-s + (−2.26 − 5.95i)12-s − 2.22·13-s + (−4.19 + 5.75i)15-s + (2.98 − 2.66i)16-s − 2.52·17-s + (1.83 + 9.94i)18-s + 5.21·19-s + ⋯ |
L(s) = 1 | + (−0.180 − 0.983i)2-s + 1.83i·3-s + (−0.934 + 0.355i)4-s + (0.808 + 0.588i)5-s + (1.80 − 0.332i)6-s + (0.519 + 0.854i)8-s − 2.38·9-s + (0.432 − 0.901i)10-s + 1.36i·11-s + (−0.654 − 1.71i)12-s − 0.617·13-s + (−1.08 + 1.48i)15-s + (0.746 − 0.665i)16-s − 0.613·17-s + (0.431 + 2.34i)18-s + 1.19·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(−0.772−0.634i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(−0.772−0.634i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
−0.772−0.634i
|
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), −0.772−0.634i)
|
Particular Values
L(1) |
≈ |
0.365235+1.02067i |
L(21) |
≈ |
0.365235+1.02067i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.255+1.39i)T |
| 5 | 1+(−1.80−1.31i)T |
| 7 | 1 |
good | 3 | 1−3.18iT−3T2 |
| 11 | 1−4.51iT−11T2 |
| 13 | 1+2.22T+13T2 |
| 17 | 1+2.52T+17T2 |
| 19 | 1−5.21T+19T2 |
| 23 | 1+1.71T+23T2 |
| 29 | 1+2.31T+29T2 |
| 31 | 1−4.62T+31T2 |
| 37 | 1+0.336iT−37T2 |
| 41 | 1+3.28iT−41T2 |
| 43 | 1+6.66T+43T2 |
| 47 | 1−1.44iT−47T2 |
| 53 | 1+10.0iT−53T2 |
| 59 | 1−3.20T+59T2 |
| 61 | 1+6.05iT−61T2 |
| 67 | 1+11.1T+67T2 |
| 71 | 1−9.15iT−71T2 |
| 73 | 1−3.24T+73T2 |
| 79 | 1−14.2iT−79T2 |
| 83 | 1−11.3iT−83T2 |
| 89 | 1−15.2iT−89T2 |
| 97 | 1+4.49T+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
−10.02862597443009280551667127119, −9.806975217189317727167841708321, −9.297636765261983243905970229008, −8.236167688832109010139482244193, −6.97254050467073586850103181796, −5.48703857837386350260505643453, −4.89521121400578764472089657100, −4.02924184458937410091243964702, −3.03874772670188949370859491209, −2.13040192019643674202094700200,
0.53528947054129035915779256045, 1.59451105614399551545037849650, 3.00319059933365154291657929075, 4.86412184340991253084314728387, 5.85351906682085150728597753197, 6.19041018277654459879101738388, 7.16721496021117492440773687690, 7.87983203544855945617906784320, 8.623885199763550222052709348205, 9.207741657735590922543838178959