L(s) = 1 | + (−0.439 − 2.49i)2-s + (−4.14 + 1.50i)4-s + (−0.358 − 0.300i)5-s + (3.03 + 1.10i)7-s + (3.05 + 5.28i)8-s + (−0.592 + 1.02i)10-s + (2.37 − 1.99i)11-s + (−0.379 + 2.15i)13-s + (1.41 − 8.04i)14-s + (5.08 − 4.26i)16-s + (1.5 − 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (1.93 + 0.705i)20-s + (−6.02 − 5.05i)22-s + (5.73 − 2.08i)23-s + ⋯ |
L(s) = 1 | + (−0.310 − 1.76i)2-s + (−2.07 + 0.754i)4-s + (−0.160 − 0.134i)5-s + (1.14 + 0.417i)7-s + (1.07 + 1.86i)8-s + (−0.187 + 0.324i)10-s + (0.717 − 0.601i)11-s + (−0.105 + 0.596i)13-s + (0.379 − 2.15i)14-s + (1.27 − 1.06i)16-s + (0.363 − 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (0.433 + 0.157i)20-s + (−1.28 − 1.07i)22-s + (1.19 − 0.435i)23-s + ⋯ |
Λ(s)=(=(729s/2ΓC(s)L(s)(−0.835+0.549i)Λ(2−s)
Λ(s)=(=(729s/2ΓC(s+1/2)L(s)(−0.835+0.549i)Λ(1−s)
Degree: |
2 |
Conductor: |
729
= 36
|
Sign: |
−0.835+0.549i
|
Analytic conductor: |
5.82109 |
Root analytic conductor: |
2.41269 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ729(82,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 729, ( :1/2), −0.835+0.549i)
|
Particular Values
L(1) |
≈ |
0.366011−1.22256i |
L(21) |
≈ |
0.366011−1.22256i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(0.439+2.49i)T+(−1.87+0.684i)T2 |
| 5 | 1+(0.358+0.300i)T+(0.868+4.92i)T2 |
| 7 | 1+(−3.03−1.10i)T+(5.36+4.49i)T2 |
| 11 | 1+(−2.37+1.99i)T+(1.91−10.8i)T2 |
| 13 | 1+(0.379−2.15i)T+(−12.2−4.44i)T2 |
| 17 | 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.0209+0.0362i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−5.73+2.08i)T+(17.6−14.7i)T2 |
| 29 | 1+(1.14+6.47i)T+(−27.2+9.91i)T2 |
| 31 | 1+(−5.85+2.12i)T+(23.7−19.9i)T2 |
| 37 | 1+(1.79−3.11i)T+(−18.5−32.0i)T2 |
| 41 | 1+(1.33−7.58i)T+(−38.5−14.0i)T2 |
| 43 | 1+(0.450−0.378i)T+(7.46−42.3i)T2 |
| 47 | 1+(9.07+3.30i)T+(36.0+30.2i)T2 |
| 53 | 1+4.95T+53T2 |
| 59 | 1+(−6.53−5.48i)T+(10.2+58.1i)T2 |
| 61 | 1+(−1.19−0.433i)T+(46.7+39.2i)T2 |
| 67 | 1+(−1.73+9.85i)T+(−62.9−22.9i)T2 |
| 71 | 1+(−5.91+10.2i)T+(−35.5−61.4i)T2 |
| 73 | 1+(−4.11−7.13i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−1.91−10.8i)T+(−74.2+27.0i)T2 |
| 83 | 1+(−0.262−1.48i)T+(−77.9+28.3i)T2 |
| 89 | 1+(7.93+13.7i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−14.2+11.9i)T+(16.8−95.5i)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
−10.09967957898776803023951041496, −9.403120570028869218356748571413, −8.536455623366138002564135392945, −8.031434216769767004403874556081, −6.47860926988457701863563141083, −4.97162961189259660385637688532, −4.34748643567407616672423956551, −3.14360952021820085942727265727, −2.07359958268742720250439342234, −0.911335230770448958832362849511,
1.32792992481616282169851449775, 3.68136969772610726992969723728, 4.85536853846760588303286308595, 5.38271481210261889698086677718, 6.62874843009933123106824394541, 7.27810161903010871777505816521, 7.962419012127302454940843398799, 8.713208173875048968228613029030, 9.544162304236430462267243599961, 10.53529802184576089599559980535