L(s) = 1 | − 3i·3-s + (2 − i)5-s − 6·9-s + 3·11-s − i·13-s + (−3 − 6i)15-s − 5i·17-s − 8·19-s + 2i·23-s + (3 − 4i)25-s + 9i·27-s + 29-s + 2·31-s − 9i·33-s − 10i·37-s + ⋯ |
L(s) = 1 | − 1.73i·3-s + (0.894 − 0.447i)5-s − 2·9-s + 0.904·11-s − 0.277i·13-s + (−0.774 − 1.54i)15-s − 1.21i·17-s − 1.83·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s + 1.73i·27-s + 0.185·29-s + 0.359·31-s − 1.56i·33-s − 1.64i·37-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(−0.894+0.447i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(−0.894+0.447i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
−0.894+0.447i
|
Analytic conductor: |
7.82533 |
Root analytic conductor: |
2.79738 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ980(589,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 980, ( :1/2), −0.894+0.447i)
|
Particular Values
L(1) |
≈ |
0.384578−1.62909i |
L(21) |
≈ |
0.384578−1.62909i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(−2+i)T |
| 7 | 1 |
good | 3 | 1+3iT−3T2 |
| 11 | 1−3T+11T2 |
| 13 | 1+iT−13T2 |
| 17 | 1+5iT−17T2 |
| 19 | 1+8T+19T2 |
| 23 | 1−2iT−23T2 |
| 29 | 1−T+29T2 |
| 31 | 1−2T+31T2 |
| 37 | 1+10iT−37T2 |
| 41 | 1−6T+41T2 |
| 43 | 1+4iT−43T2 |
| 47 | 1−11iT−47T2 |
| 53 | 1−6iT−53T2 |
| 59 | 1+10T+59T2 |
| 61 | 1+61T2 |
| 67 | 1−10iT−67T2 |
| 71 | 1+71T2 |
| 73 | 1−10iT−73T2 |
| 79 | 1−7T+79T2 |
| 83 | 1+12iT−83T2 |
| 89 | 1−8T+89T2 |
| 97 | 1−3iT−97T2 |
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show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.308647782440550662308877980361, −8.839989837474824533469769095291, −7.85331663001520225087384247214, −7.03991450957754779616423359108, −6.27392318827859543239492030933, −5.70687099983519711550228008734, −4.41093566146988608438378054487, −2.71340007281772235735847648629, −1.86820668411512967594478598834, −0.77182193178669602856374884423,
2.01841951785734878764055362974, 3.31062373178681191280085470489, 4.18923555886163601576632259182, 4.92756198997538272549689692150, 6.17872031279806468540302374260, 6.48267042328027339439777400728, 8.261941328595009393414374843364, 8.935894304999737881634882744772, 9.622560021151432896492148975476, 10.41335474582043844198113590174