L(s) = 1 | + (1.20 − 1.01i)2-s + (0.0853 − 0.483i)4-s + (1.57 − 0.574i)5-s + (0.482 + 2.73i)7-s + (1.19 + 2.06i)8-s + (1.32 − 2.29i)10-s + (−3.90 − 1.41i)11-s + (5.26 + 4.41i)13-s + (3.36 + 2.81i)14-s + (4.45 + 1.62i)16-s + (0.488 − 0.845i)17-s + (−1.34 − 2.32i)19-s + (−0.143 − 0.812i)20-s + (−6.15 + 2.24i)22-s + (−0.280 + 1.58i)23-s + ⋯ |
L(s) = 1 | + (0.854 − 0.717i)2-s + (0.0426 − 0.241i)4-s + (0.705 − 0.256i)5-s + (0.182 + 1.03i)7-s + (0.420 + 0.729i)8-s + (0.418 − 0.725i)10-s + (−1.17 − 0.428i)11-s + (1.46 + 1.22i)13-s + (0.898 + 0.753i)14-s + (1.11 + 0.405i)16-s + (0.118 − 0.205i)17-s + (−0.308 − 0.533i)19-s + (−0.0320 − 0.181i)20-s + (−1.31 + 0.477i)22-s + (−0.0584 + 0.331i)23-s + ⋯ |
Λ(s)=(=(729s/2ΓC(s)L(s)(0.993+0.116i)Λ(2−s)
Λ(s)=(=(729s/2ΓC(s+1/2)L(s)(0.993+0.116i)Λ(1−s)
Degree: |
2 |
Conductor: |
729
= 36
|
Sign: |
0.993+0.116i
|
Analytic conductor: |
5.82109 |
Root analytic conductor: |
2.41269 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ729(325,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 729, ( :1/2), 0.993+0.116i)
|
Particular Values
L(1) |
≈ |
2.69653−0.157055i |
L(21) |
≈ |
2.69653−0.157055i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(−1.20+1.01i)T+(0.347−1.96i)T2 |
| 5 | 1+(−1.57+0.574i)T+(3.83−3.21i)T2 |
| 7 | 1+(−0.482−2.73i)T+(−6.57+2.39i)T2 |
| 11 | 1+(3.90+1.41i)T+(8.42+7.07i)T2 |
| 13 | 1+(−5.26−4.41i)T+(2.25+12.8i)T2 |
| 17 | 1+(−0.488+0.845i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.34+2.32i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.280−1.58i)T+(−21.6−7.86i)T2 |
| 29 | 1+(−6.30+5.28i)T+(5.03−28.5i)T2 |
| 31 | 1+(−0.181+1.02i)T+(−29.1−10.6i)T2 |
| 37 | 1+(−0.654+1.13i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−3.71−3.11i)T+(7.11+40.3i)T2 |
| 43 | 1+(9.24+3.36i)T+(32.9+27.6i)T2 |
| 47 | 1+(2.17+12.3i)T+(−44.1+16.0i)T2 |
| 53 | 1+7.34T+53T2 |
| 59 | 1+(−8.50+3.09i)T+(45.1−37.9i)T2 |
| 61 | 1+(0.223+1.26i)T+(−57.3+20.8i)T2 |
| 67 | 1+(3.55+2.98i)T+(11.6+65.9i)T2 |
| 71 | 1+(−2.81+4.87i)T+(−35.5−61.4i)T2 |
| 73 | 1+(−2.28−3.95i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−3.56+2.99i)T+(13.7−77.7i)T2 |
| 83 | 1+(4.41−3.70i)T+(14.4−81.7i)T2 |
| 89 | 1+(−2.27−3.93i)T+(−44.5+77.0i)T2 |
| 97 | 1+(8.05+2.93i)T+(74.3+62.3i)T2 |
show more | |
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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
−10.64951790727712446526183225068, −9.586663975902543337051108741323, −8.629906550504637773620123745772, −8.095519029831740829739320770718, −6.50681592971676960649775989423, −5.61388287960138199912892839790, −4.95717705187961298713356025618, −3.79803093107750623640888962079, −2.63851372297280956250664876446, −1.80829686228648276669424081989,
1.24268772299173432549447153068, 3.04648323457374325379816934716, 4.15492682314594236007902950467, 5.12586927324043773552328793465, 5.95264552498003609574194772098, 6.62892563385373133566273416650, 7.68287039531845740402759558385, 8.329447247023790342705912743989, 9.906168734344787103045661698861, 10.43071038208761870896436438286