L(s) = 1 | − 3.50·2-s + (−2.88 − 0.822i)3-s + 8.29·4-s + (10.1 + 2.88i)6-s + 2.64i·7-s − 15.0·8-s + (7.64 + 4.74i)9-s + 7.01i·11-s + (−23.9 − 6.82i)12-s + 11.6i·13-s − 9.27i·14-s + 19.5·16-s + 4.52·17-s + (−26.8 − 16.6i)18-s − 16.2·19-s + ⋯ |
L(s) = 1 | − 1.75·2-s + (−0.961 − 0.274i)3-s + 2.07·4-s + (1.68 + 0.480i)6-s + 0.377i·7-s − 1.88·8-s + (0.849 + 0.527i)9-s + 0.637i·11-s + (−1.99 − 0.568i)12-s + 0.895i·13-s − 0.662i·14-s + 1.22·16-s + 0.266·17-s + (−1.48 − 0.924i)18-s − 0.854·19-s + ⋯ |
Λ(s)=(=(525s/2ΓC(s)L(s)(−0.737−0.675i)Λ(3−s)
Λ(s)=(=(525s/2ΓC(s+1)L(s)(−0.737−0.675i)Λ(1−s)
Degree: |
2 |
Conductor: |
525
= 3⋅52⋅7
|
Sign: |
−0.737−0.675i
|
Analytic conductor: |
14.3052 |
Root analytic conductor: |
3.78222 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ525(449,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 525, ( :1), −0.737−0.675i)
|
Particular Values
L(23) |
≈ |
0.2509021794 |
L(21) |
≈ |
0.2509021794 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(2.88+0.822i)T |
| 5 | 1 |
| 7 | 1−2.64iT |
good | 2 | 1+3.50T+4T2 |
| 11 | 1−7.01iT−121T2 |
| 13 | 1−11.6iT−169T2 |
| 17 | 1−4.52T+289T2 |
| 19 | 1+16.2T+361T2 |
| 23 | 1−25.5T+529T2 |
| 29 | 1−9.49iT−841T2 |
| 31 | 1−28.7T+961T2 |
| 37 | 1+33.0iT−1.36e3T2 |
| 41 | 1+67.1iT−1.68e3T2 |
| 43 | 1−24.1iT−1.84e3T2 |
| 47 | 1+33.0T+2.20e3T2 |
| 53 | 1−15.1T+2.80e3T2 |
| 59 | 1−92.3iT−3.48e3T2 |
| 61 | 1+57.5T+3.72e3T2 |
| 67 | 1−15.1iT−4.48e3T2 |
| 71 | 1−70.5iT−5.04e3T2 |
| 73 | 1−76.7iT−5.32e3T2 |
| 79 | 1+127.T+6.24e3T2 |
| 83 | 1+74.2T+6.88e3T2 |
| 89 | 1+127.iT−7.92e3T2 |
| 97 | 1+23.1iT−9.40e3T2 |
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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
−10.79248929326776951025585014227, −10.11609451384325214113471507073, −9.233564641713618833138052718106, −8.472454276649004073935737831114, −7.30181138622171752916101680263, −6.84359794821081223035319738105, −5.81472945787175058487773011994, −4.47797966368383683688004744060, −2.35594460415009739122921186981, −1.26644693876941901087365725537,
0.23989718834349543539577315248, 1.28365855846023892076999212243, 3.09363940805499963968038577497, 4.75569676792971331114875772475, 6.07325040311523508925414051615, 6.76933123819680012363827147818, 7.81290359052971347423357053952, 8.554603133307914506661868828484, 9.639303347457660926720929161704, 10.24653333164115489065411647573