L(s) = 1 | + (0.169 + 1.40i)2-s + (−0.416 − 0.240i)3-s + (−1.94 + 0.477i)4-s + (1.59 + 2.76i)5-s + (0.266 − 0.625i)6-s + (0.694 − 2.55i)7-s + (−0.999 − 2.64i)8-s + (−1.38 − 2.39i)9-s + (−3.61 + 2.71i)10-s + (0.800 − 1.38i)11-s + (0.923 + 0.268i)12-s − 1.38·13-s + (3.70 + 0.540i)14-s − 1.53i·15-s + (3.54 − 1.85i)16-s + (3.48 + 2.01i)17-s + ⋯ |
L(s) = 1 | + (0.120 + 0.992i)2-s + (−0.240 − 0.138i)3-s + (−0.971 + 0.238i)4-s + (0.714 + 1.23i)5-s + (0.108 − 0.255i)6-s + (0.262 − 0.964i)7-s + (−0.353 − 0.935i)8-s + (−0.461 − 0.799i)9-s + (−1.14 + 0.857i)10-s + (0.241 − 0.418i)11-s + (0.266 + 0.0774i)12-s − 0.385·13-s + (0.989 + 0.144i)14-s − 0.396i·15-s + (0.886 − 0.463i)16-s + (0.845 + 0.488i)17-s + ⋯ |
Λ(s)=(=(56s/2ΓC(s)L(s)(0.296−0.955i)Λ(2−s)
Λ(s)=(=(56s/2ΓC(s+1/2)L(s)(0.296−0.955i)Λ(1−s)
Degree: |
2 |
Conductor: |
56
= 23⋅7
|
Sign: |
0.296−0.955i
|
Analytic conductor: |
0.447162 |
Root analytic conductor: |
0.668701 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ56(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 56, ( :1/2), 0.296−0.955i)
|
Particular Values
L(1) |
≈ |
0.679297+0.500424i |
L(21) |
≈ |
0.679297+0.500424i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.169−1.40i)T |
| 7 | 1+(−0.694+2.55i)T |
good | 3 | 1+(0.416+0.240i)T+(1.5+2.59i)T2 |
| 5 | 1+(−1.59−2.76i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−0.800+1.38i)T+(−5.5−9.52i)T2 |
| 13 | 1+1.38T+13T2 |
| 17 | 1+(−3.48−2.01i)T+(8.5+14.7i)T2 |
| 19 | 1+(4.56−2.63i)T+(9.5−16.4i)T2 |
| 23 | 1+(3.83−2.21i)T+(11.5−19.9i)T2 |
| 29 | 1+5.10iT−29T2 |
| 31 | 1+(0.0579−0.100i)T+(−15.5−26.8i)T2 |
| 37 | 1+(4.63−2.67i)T+(18.5−32.0i)T2 |
| 41 | 1+4.21iT−41T2 |
| 43 | 1+43T2 |
| 47 | 1+(−5.05−8.76i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−6.13−3.54i)T+(26.5+45.8i)T2 |
| 59 | 1+(4.38+2.53i)T+(29.5+51.0i)T2 |
| 61 | 1+(4.21+7.29i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−5.01+8.69i)T+(−33.5−58.0i)T2 |
| 71 | 1−5.29iT−71T2 |
| 73 | 1+(−9.30−5.37i)T+(36.5+63.2i)T2 |
| 79 | 1+(−10.3+5.96i)T+(39.5−68.4i)T2 |
| 83 | 1−14.9iT−83T2 |
| 89 | 1+(−1.5+0.866i)T+(44.5−77.0i)T2 |
| 97 | 1+2.87iT−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
−15.26471919884032658692676230840, −14.32231904970114104722852578902, −13.86181221046588967478742979815, −12.31260370706290523786878010874, −10.70554870643721000316842646601, −9.673751035589436658578175764006, −7.977218556795374272567443228196, −6.69475817690909313554539557738, −5.88071359817964704063112862272, −3.71851100473250361583582482992,
2.15983832251087429689402678176, 4.78194298745032568164569571545, 5.57637555101040971646264820839, 8.409918306769938684211647501948, 9.254000141675969425264204001226, 10.45007910362127310451250615853, 11.87410335070880401719929512203, 12.56942797711840506829046901143, 13.66817660180281782305695386624, 14.78446360208411867330720473170