L(s) = 1 | + (0.388 + 0.673i)2-s + (0.697 − 1.20i)4-s + (1.18 − 2.05i)5-s + (1.25 + 2.16i)7-s + 2.64·8-s + 1.84·10-s + (1.57 + 2.72i)11-s + (−0.668 + 1.15i)13-s + (−0.972 + 1.68i)14-s + (−0.368 − 0.637i)16-s − 6.27·17-s + 8.06·19-s + (−1.65 − 2.87i)20-s + (−1.22 + 2.11i)22-s + (2.02 − 3.51i)23-s + ⋯ |
L(s) = 1 | + (0.274 + 0.476i)2-s + (0.348 − 0.604i)4-s + (0.531 − 0.920i)5-s + (0.472 + 0.818i)7-s + 0.933·8-s + 0.584·10-s + (0.473 + 0.820i)11-s + (−0.185 + 0.321i)13-s + (−0.259 + 0.450i)14-s + (−0.0920 − 0.159i)16-s − 1.52·17-s + 1.85·19-s + (−0.370 − 0.642i)20-s + (−0.260 + 0.451i)22-s + (0.422 − 0.732i)23-s + ⋯ |
Λ(s)=(=(729s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(729s/2ΓC(s+1/2)L(s)Λ(1−s)
Degree: |
2 |
Conductor: |
729
= 36
|
Sign: |
1
|
Analytic conductor: |
5.82109 |
Root analytic conductor: |
2.41269 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ729(244,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 729, ( :1/2), 1)
|
Particular Values
L(1) |
≈ |
2.32882 |
L(21) |
≈ |
2.32882 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(−0.388−0.673i)T+(−1+1.73i)T2 |
| 5 | 1+(−1.18+2.05i)T+(−2.5−4.33i)T2 |
| 7 | 1+(−1.25−2.16i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−1.57−2.72i)T+(−5.5+9.52i)T2 |
| 13 | 1+(0.668−1.15i)T+(−6.5−11.2i)T2 |
| 17 | 1+6.27T+17T2 |
| 19 | 1−8.06T+19T2 |
| 23 | 1+(−2.02+3.51i)T+(−11.5−19.9i)T2 |
| 29 | 1+(4.64+8.04i)T+(−14.5+25.1i)T2 |
| 31 | 1+(1.41−2.45i)T+(−15.5−26.8i)T2 |
| 37 | 1−5.53T+37T2 |
| 41 | 1+(3.55−6.15i)T+(−20.5−35.5i)T2 |
| 43 | 1+(1.16+2.02i)T+(−21.5+37.2i)T2 |
| 47 | 1+(2.30+3.99i)T+(−23.5+40.7i)T2 |
| 53 | 1+0.135T+53T2 |
| 59 | 1+(1.99−3.46i)T+(−29.5−51.0i)T2 |
| 61 | 1+(0.170+0.296i)T+(−30.5+52.8i)T2 |
| 67 | 1+(5.06−8.76i)T+(−33.5−58.0i)T2 |
| 71 | 1+8.19T+71T2 |
| 73 | 1+12.3T+73T2 |
| 79 | 1+(2.04+3.53i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−0.456−0.790i)T+(−41.5+71.8i)T2 |
| 89 | 1−3.72T+89T2 |
| 97 | 1+(−2.99−5.19i)T+(−48.5+84.0i)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
−10.22374936551372005808025845885, −9.390575623874343473979726831055, −8.879496355611182204470469227716, −7.64751552286770437342862948403, −6.76374724687261232175710179859, −5.82807747654853387927488908603, −5.04553970700473545110208740260, −4.43231052278617749852646212338, −2.35949029598725030653252059206, −1.41892294634490554588762506530,
1.53965201446149950478751002517, 2.88879389064029771475394957812, 3.58827201348560494321467195076, 4.76254488527853835085174320210, 6.01708455301529318495399845725, 7.14473376277541375493269445153, 7.45439018504141444559277047007, 8.715031845666089433869888043123, 9.712141844987246544041214406452, 10.82714480498227681716694696944