L(s) = 1 | + (0.939 − 1.62i)2-s + (−1.70 − 0.300i)3-s + (−0.766 − 1.32i)4-s + (−1.43 − 2.49i)5-s + (−2.09 + 2.49i)6-s + (0.326 − 0.565i)7-s + 0.879·8-s + (2.81 + 1.02i)9-s − 5.41·10-s + (0.5 − 0.866i)11-s + (0.907 + 2.49i)12-s + (3.37 + 5.85i)13-s + (−0.613 − 1.06i)14-s + (1.70 + 4.68i)15-s + (2.35 − 4.08i)16-s + 0.184·17-s + ⋯ |
L(s) = 1 | + (0.664 − 1.15i)2-s + (−0.984 − 0.173i)3-s + (−0.383 − 0.663i)4-s + (−0.643 − 1.11i)5-s + (−0.854 + 1.01i)6-s + (0.123 − 0.213i)7-s + 0.310·8-s + (0.939 + 0.342i)9-s − 1.71·10-s + (0.150 − 0.261i)11-s + (0.262 + 0.719i)12-s + (0.937 + 1.62i)13-s + (−0.163 − 0.283i)14-s + (0.440 + 1.21i)15-s + (0.589 − 1.02i)16-s + 0.0448·17-s + ⋯ |
Λ(s)=(=(99s/2ΓC(s)L(s)(−0.500+0.866i)Λ(2−s)
Λ(s)=(=(99s/2ΓC(s+1/2)L(s)(−0.500+0.866i)Λ(1−s)
Degree: |
2 |
Conductor: |
99
= 32⋅11
|
Sign: |
−0.500+0.866i
|
Analytic conductor: |
0.790518 |
Root analytic conductor: |
0.889111 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ99(34,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 99, ( :1/2), −0.500+0.866i)
|
Particular Values
L(1) |
≈ |
0.518567−0.898185i |
L(21) |
≈ |
0.518567−0.898185i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.70+0.300i)T |
| 11 | 1+(−0.5+0.866i)T |
good | 2 | 1+(−0.939+1.62i)T+(−1−1.73i)T2 |
| 5 | 1+(1.43+2.49i)T+(−2.5+4.33i)T2 |
| 7 | 1+(−0.326+0.565i)T+(−3.5−6.06i)T2 |
| 13 | 1+(−3.37−5.85i)T+(−6.5+11.2i)T2 |
| 17 | 1−0.184T+17T2 |
| 19 | 1+5.22T+19T2 |
| 23 | 1+(−1.59−2.75i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−2.01+3.48i)T+(−14.5−25.1i)T2 |
| 31 | 1+(0.553+0.957i)T+(−15.5+26.8i)T2 |
| 37 | 1−0.106T+37T2 |
| 41 | 1+(−2.80−4.86i)T+(−20.5+35.5i)T2 |
| 43 | 1+(1.92−3.34i)T+(−21.5−37.2i)T2 |
| 47 | 1+(6.00−10.3i)T+(−23.5−40.7i)T2 |
| 53 | 1+10.0T+53T2 |
| 59 | 1+(5.27+9.14i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−3.67+6.36i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−5.90−10.2i)T+(−33.5+58.0i)T2 |
| 71 | 1+2.47T+71T2 |
| 73 | 1−10.4T+73T2 |
| 79 | 1+(−0.733+1.27i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−0.520+0.902i)T+(−41.5−71.8i)T2 |
| 89 | 1+3.01T+89T2 |
| 97 | 1+(−2.86+4.97i)T+(−48.5−84.0i)T2 |
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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
−13.06057713531468769412059313647, −12.43760275348914562073493384079, −11.40381364776942494347732484418, −11.08948993537939147894747004982, −9.499955869669467671586087336823, −8.033488627298117695249270513392, −6.39750311496769766351948640263, −4.72213850669722601074317444081, −4.07484953111814333481348863115, −1.43255679275398639033860616590,
3.73666614660862836087454104850, 5.17843851421376892230602417199, 6.28886996829218224912112058240, 7.07013772828168894407102789621, 8.266052948927218701638265756799, 10.51179848334022285439567020510, 10.86059941714504891459072775709, 12.31907917792042465022701315280, 13.29674243050505664512120908319, 14.75648955819797762141413907904