L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.01 + 0.272i)3-s + (−0.499 + 0.866i)4-s + (−0.272 − 1.01i)6-s + (−0.524 − 0.303i)7-s + 0.999·8-s + (−1.63 − 0.944i)9-s + (1.67 − 6.24i)11-s + (−0.745 + 0.745i)12-s + (−3.55 + 0.572i)13-s + 0.606i·14-s + (−0.5 − 0.866i)16-s + (0.267 + 0.996i)17-s + 1.88i·18-s + (0.896 − 0.240i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.587 + 0.157i)3-s + (−0.249 + 0.433i)4-s + (−0.111 − 0.415i)6-s + (−0.198 − 0.114i)7-s + 0.353·8-s + (−0.545 − 0.314i)9-s + (0.504 − 1.88i)11-s + (−0.215 + 0.215i)12-s + (−0.987 + 0.158i)13-s + 0.162i·14-s + (−0.125 − 0.216i)16-s + (0.0647 + 0.241i)17-s + 0.445i·18-s + (0.205 − 0.0551i)19-s + ⋯ |
Λ(s)=(=(650s/2ΓC(s)L(s)(−0.444+0.895i)Λ(2−s)
Λ(s)=(=(650s/2ΓC(s+1/2)L(s)(−0.444+0.895i)Λ(1−s)
Degree: |
2 |
Conductor: |
650
= 2⋅52⋅13
|
Sign: |
−0.444+0.895i
|
Analytic conductor: |
5.19027 |
Root analytic conductor: |
2.27821 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ650(457,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 650, ( :1/2), −0.444+0.895i)
|
Particular Values
L(1) |
≈ |
0.601390−0.969816i |
L(21) |
≈ |
0.601390−0.969816i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.5+0.866i)T |
| 5 | 1 |
| 13 | 1+(3.55−0.572i)T |
good | 3 | 1+(−1.01−0.272i)T+(2.59+1.5i)T2 |
| 7 | 1+(0.524+0.303i)T+(3.5+6.06i)T2 |
| 11 | 1+(−1.67+6.24i)T+(−9.52−5.5i)T2 |
| 17 | 1+(−0.267−0.996i)T+(−14.7+8.5i)T2 |
| 19 | 1+(−0.896+0.240i)T+(16.4−9.5i)T2 |
| 23 | 1+(−1.31+4.90i)T+(−19.9−11.5i)T2 |
| 29 | 1+(−2.65+1.53i)T+(14.5−25.1i)T2 |
| 31 | 1+(−7.06+7.06i)T−31iT2 |
| 37 | 1+(0.0372−0.0214i)T+(18.5−32.0i)T2 |
| 41 | 1+(−1.75−0.471i)T+(35.5+20.5i)T2 |
| 43 | 1+(6.78−1.81i)T+(37.2−21.5i)T2 |
| 47 | 1−7.31iT−47T2 |
| 53 | 1+(−3.80+3.80i)T−53iT2 |
| 59 | 1+(1.52+5.68i)T+(−51.0+29.5i)T2 |
| 61 | 1+(1.28−2.21i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.28−5.69i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−2.71−10.1i)T+(−61.4+35.5i)T2 |
| 73 | 1+2.04T+73T2 |
| 79 | 1−4.09iT−79T2 |
| 83 | 1−7.40iT−83T2 |
| 89 | 1+(14.6+3.91i)T+(77.0+44.5i)T2 |
| 97 | 1+(−8.80+15.2i)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
−10.10195285144228253817554288348, −9.469883299697114706470980542809, −8.489100783591493068966845483811, −8.173987813272685575016436685092, −6.72564488623917513992764687741, −5.79691521323199526242315529519, −4.36606396526152303415887995322, −3.30017404241711820895254420507, −2.56686346745853954549385517983, −0.64369963136564601401032463651,
1.79197391499946903380810751165, 3.02168733563119695824954787989, 4.58351005991489998382710647017, 5.34137437610281156195506823443, 6.72592440924782611889788076068, 7.33384851771457640626475314297, 8.118434728683662550751089589540, 9.136539593115415823964082092136, 9.707905245438556284702031378220, 10.49245338457614352545588482899