L(s) = 1 | + (0.0982 + 1.41i)2-s − 0.662·3-s + (−1.98 + 0.277i)4-s − i·5-s + (−0.0650 − 0.934i)6-s + (−0.585 − 2.76i)8-s − 2.56·9-s + (1.41 − 0.0982i)10-s − 3.61i·11-s + (1.31 − 0.183i)12-s + 5.83i·13-s + 0.662i·15-s + (3.84 − 1.09i)16-s + 1.36i·17-s + (−0.251 − 3.61i)18-s + 4.09·19-s + ⋯ |
L(s) = 1 | + (0.0694 + 0.997i)2-s − 0.382·3-s + (−0.990 + 0.138i)4-s − 0.447i·5-s + (−0.0265 − 0.381i)6-s + (−0.207 − 0.978i)8-s − 0.853·9-s + (0.446 − 0.0310i)10-s − 1.08i·11-s + (0.378 − 0.0530i)12-s + 1.61i·13-s + 0.171i·15-s + (0.961 − 0.274i)16-s + 0.331i·17-s + (−0.0593 − 0.851i)18-s + 0.939·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.543−0.839i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.543−0.839i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.543−0.839i
|
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), 0.543−0.839i)
|
Particular Values
L(1) |
≈ |
1.01250+0.550604i |
L(21) |
≈ |
1.01250+0.550604i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.0982−1.41i)T |
| 5 | 1+iT |
| 7 | 1 |
good | 3 | 1+0.662T+3T2 |
| 11 | 1+3.61iT−11T2 |
| 13 | 1−5.83iT−13T2 |
| 17 | 1−1.36iT−17T2 |
| 19 | 1−4.09T+19T2 |
| 23 | 1+3.24iT−23T2 |
| 29 | 1−5.19T+29T2 |
| 31 | 1−8.86T+31T2 |
| 37 | 1−10.7T+37T2 |
| 41 | 1+0.832iT−41T2 |
| 43 | 1−3.10iT−43T2 |
| 47 | 1−6.89T+47T2 |
| 53 | 1+7.41T+53T2 |
| 59 | 1+7.47T+59T2 |
| 61 | 1−1.48iT−61T2 |
| 67 | 1+2.53iT−67T2 |
| 71 | 1+3.52iT−71T2 |
| 73 | 1−5.16iT−73T2 |
| 79 | 1+11.3iT−79T2 |
| 83 | 1−6.49T+83T2 |
| 89 | 1−9.39iT−89T2 |
| 97 | 1+0.343iT−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
−9.866003800276742497857474468523, −9.021286891217894165737965485876, −8.477709469942706503498909496746, −7.66963930167735805265067247749, −6.33562720281422423908432659348, −6.16320983634546927950327059566, −4.98265523066639655396110693941, −4.26913829550570858415638577945, −2.95794383821513887880965228642, −0.851672199187797611040417859270,
0.866690878850131891519749437811, 2.58275543880072277579566483207, 3.18812496556779823014574865856, 4.56301461383469073931401288837, 5.36604434017344638875691969258, 6.20085523775478189407557152200, 7.57280438259178861360793229185, 8.217785925043430178260648715829, 9.387832579491188085264425486875, 10.04324212873773308371455617112