L(s) = 1 | + (1.85 + 1.85i)2-s + (0.866 − 0.5i)3-s + 4.91i·4-s + (0.502 + 0.134i)5-s + (2.53 + 0.680i)6-s + (−1.89 − 1.84i)7-s + (−5.41 + 5.41i)8-s + (0.499 − 0.866i)9-s + (0.684 + 1.18i)10-s + (−1.93 − 0.517i)11-s + (2.45 + 4.25i)12-s + (2.40 + 2.68i)13-s + (−0.0921 − 6.95i)14-s + (0.502 − 0.134i)15-s − 10.3·16-s + 5.09·17-s + ⋯ |
L(s) = 1 | + (1.31 + 1.31i)2-s + (0.499 − 0.288i)3-s + 2.45i·4-s + (0.224 + 0.0602i)5-s + (1.03 + 0.277i)6-s + (−0.716 − 0.697i)7-s + (−1.91 + 1.91i)8-s + (0.166 − 0.288i)9-s + (0.216 + 0.374i)10-s + (−0.582 − 0.156i)11-s + (0.709 + 1.22i)12-s + (0.667 + 0.744i)13-s + (−0.0246 − 1.85i)14-s + (0.129 − 0.0347i)15-s − 2.57·16-s + 1.23·17-s + ⋯ |
Λ(s)=(=(273s/2ΓC(s)L(s)(−0.110−0.993i)Λ(2−s)
Λ(s)=(=(273s/2ΓC(s+1/2)L(s)(−0.110−0.993i)Λ(1−s)
Degree: |
2 |
Conductor: |
273
= 3⋅7⋅13
|
Sign: |
−0.110−0.993i
|
Analytic conductor: |
2.17991 |
Root analytic conductor: |
1.47645 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ273(241,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 273, ( :1/2), −0.110−0.993i)
|
Particular Values
L(1) |
≈ |
1.76119+1.96876i |
L(21) |
≈ |
1.76119+1.96876i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.866+0.5i)T |
| 7 | 1+(1.89+1.84i)T |
| 13 | 1+(−2.40−2.68i)T |
good | 2 | 1+(−1.85−1.85i)T+2iT2 |
| 5 | 1+(−0.502−0.134i)T+(4.33+2.5i)T2 |
| 11 | 1+(1.93+0.517i)T+(9.52+5.5i)T2 |
| 17 | 1−5.09T+17T2 |
| 19 | 1+(1.86+6.96i)T+(−16.4+9.5i)T2 |
| 23 | 1+4.78iT−23T2 |
| 29 | 1+(1.35−2.34i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−0.867−3.23i)T+(−26.8+15.5i)T2 |
| 37 | 1+(5.25−5.25i)T−37iT2 |
| 41 | 1+(−0.938−3.50i)T+(−35.5+20.5i)T2 |
| 43 | 1+(3.62−2.09i)T+(21.5−37.2i)T2 |
| 47 | 1+(−0.0215+0.0803i)T+(−40.7−23.5i)T2 |
| 53 | 1+(−6.85+11.8i)T+(−26.5−45.8i)T2 |
| 59 | 1+(3.91+3.91i)T+59iT2 |
| 61 | 1+(0.652+0.376i)T+(30.5+52.8i)T2 |
| 67 | 1+(2.81−10.4i)T+(−58.0−33.5i)T2 |
| 71 | 1+(1.33−5.00i)T+(−61.4−35.5i)T2 |
| 73 | 1+(−2.05+0.551i)T+(63.2−36.5i)T2 |
| 79 | 1+(−6.17−10.6i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−7.09+7.09i)T−83iT2 |
| 89 | 1+(9.64+9.64i)T+89iT2 |
| 97 | 1+(3.28+0.880i)T+(84.0+48.5i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.74485551592370900194132972208, −11.64141264876491979224224495487, −10.21086216823439901329881835299, −8.842943775132587375975980048361, −7.947526223021500531702630479489, −6.88673165287671898961379998529, −6.40085617293384239038192517744, −5.08793620619979569467294625834, −3.92443388983074872860193357384, −2.90080146957259594864366034840,
1.87847759347150946885284929564, 3.17328488885341760071016080032, 3.85747761596617113510230258920, 5.54239541314921387225821652987, 5.82991547246538118793507007726, 7.84497591658453515952439515624, 9.303050374738053039367055153789, 10.06592534349962668920262965189, 10.69882510536668582962052064434, 12.00642904156962593914008830341