L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−3.04 + 3.04i)5-s + (−0.258 − 0.965i)6-s + (−1.02 + 2.43i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−2.15 − 3.72i)10-s + (−2.01 − 0.539i)11-s + 12-s + (0.973 − 3.47i)13-s + (−2.08 − 1.62i)14-s + (1.11 − 4.15i)15-s + (0.500 + 0.866i)16-s + (−0.994 + 1.72i)17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.36 + 1.36i)5-s + (−0.105 − 0.394i)6-s + (−0.388 + 0.921i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.680 − 1.17i)10-s + (−0.606 − 0.162i)11-s + 0.288·12-s + (0.269 − 0.962i)13-s + (−0.558 − 0.434i)14-s + (0.287 − 1.07i)15-s + (0.125 + 0.216i)16-s + (−0.241 + 0.417i)17-s + ⋯ |
Λ(s)=(=(546s/2ΓC(s)L(s)(0.150+0.988i)Λ(2−s)
Λ(s)=(=(546s/2ΓC(s+1/2)L(s)(0.150+0.988i)Λ(1−s)
Degree: |
2 |
Conductor: |
546
= 2⋅3⋅7⋅13
|
Sign: |
0.150+0.988i
|
Analytic conductor: |
4.35983 |
Root analytic conductor: |
2.08802 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ546(349,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 546, ( :1/2), 0.150+0.988i)
|
Particular Values
L(1) |
≈ |
0.0911417−0.0782842i |
L(21) |
≈ |
0.0911417−0.0782842i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.258−0.965i)T |
| 3 | 1+(0.866−0.5i)T |
| 7 | 1+(1.02−2.43i)T |
| 13 | 1+(−0.973+3.47i)T |
good | 5 | 1+(3.04−3.04i)T−5iT2 |
| 11 | 1+(2.01+0.539i)T+(9.52+5.5i)T2 |
| 17 | 1+(0.994−1.72i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.11−4.15i)T+(−16.4+9.5i)T2 |
| 23 | 1+(−0.911+0.526i)T+(11.5−19.9i)T2 |
| 29 | 1+(−2.33−4.04i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−5.73+5.73i)T−31iT2 |
| 37 | 1+(−2.17−0.582i)T+(32.0+18.5i)T2 |
| 41 | 1+(9.88+2.64i)T+(35.5+20.5i)T2 |
| 43 | 1+(7.68+4.43i)T+(21.5+37.2i)T2 |
| 47 | 1+(0.833+0.833i)T+47iT2 |
| 53 | 1+6.61T+53T2 |
| 59 | 1+(−7.58+2.03i)T+(51.0−29.5i)T2 |
| 61 | 1+(9.14+5.27i)T+(30.5+52.8i)T2 |
| 67 | 1+(3.51−13.1i)T+(−58.0−33.5i)T2 |
| 71 | 1+(−3.38+0.908i)T+(61.4−35.5i)T2 |
| 73 | 1+(−4.18−4.18i)T+73iT2 |
| 79 | 1+2.85T+79T2 |
| 83 | 1+(9.59−9.59i)T−83iT2 |
| 89 | 1+(1.02−3.81i)T+(−77.0−44.5i)T2 |
| 97 | 1+(−1.29−4.83i)T+(−84.0+48.5i)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
−11.41244635924600478322705446317, −10.48832618018104624837225308512, −9.984457574396964642868343285455, −8.437085930486302453052606865573, −7.989112928327192305949435860113, −6.88249636735653475961410004222, −6.16798114254954230809024480182, −5.18542742115224681636564934820, −3.80275684497677991090633784587, −2.91549042134991527098845097036,
0.088559566094722090639835191710, 1.25921686894358984149064901107, 3.28327331613262837456743937800, 4.55056188835194569774302168456, 4.82539457252337378321316687251, 6.66393976576948100421144573997, 7.57246415738368906632291046158, 8.370955731367755625093685095553, 9.243461281375022591483644898908, 10.22476327545387284167154001343