L(s) = 1 | + (7.93 − 13.7i)5-s + (7.93 + 13.7i)11-s + 26·13-s + (−39.6 − 68.7i)17-s + (−34 + 58.8i)19-s + (23.8 − 41.2i)23-s + (−63.5 − 109. i)25-s + 253.·29-s + (−106 − 183. i)31-s + (−109 + 188. i)37-s − 396.·41-s + 260·43-s + (206. − 357. i)47-s + (−238. − 412. i)53-s + 252·55-s + ⋯ |
L(s) = 1 | + (0.709 − 1.22i)5-s + (0.217 + 0.376i)11-s + 0.554·13-s + (−0.566 − 0.980i)17-s + (−0.410 + 0.711i)19-s + (0.215 − 0.373i)23-s + (−0.508 − 0.879i)25-s + 1.62·29-s + (−0.614 − 1.06i)31-s + (−0.484 + 0.838i)37-s − 1.51·41-s + 0.922·43-s + (0.640 − 1.10i)47-s + (−0.617 − 1.06i)53-s + 0.617·55-s + ⋯ |
Λ(s)=(=(1764s/2ΓC(s)L(s)(−0.605+0.795i)Λ(4−s)
Λ(s)=(=(1764s/2ΓC(s+3/2)L(s)(−0.605+0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
1764
= 22⋅32⋅72
|
Sign: |
−0.605+0.795i
|
Analytic conductor: |
104.079 |
Root analytic conductor: |
10.2019 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1764(1549,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1764, ( :3/2), −0.605+0.795i)
|
Particular Values
L(2) |
≈ |
1.988769560 |
L(21) |
≈ |
1.988769560 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1 |
good | 5 | 1+(−7.93+13.7i)T+(−62.5−108.i)T2 |
| 11 | 1+(−7.93−13.7i)T+(−665.5+1.15e3i)T2 |
| 13 | 1−26T+2.19e3T2 |
| 17 | 1+(39.6+68.7i)T+(−2.45e3+4.25e3i)T2 |
| 19 | 1+(34−58.8i)T+(−3.42e3−5.94e3i)T2 |
| 23 | 1+(−23.8+41.2i)T+(−6.08e3−1.05e4i)T2 |
| 29 | 1−253.T+2.43e4T2 |
| 31 | 1+(106+183.i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+(109−188.i)T+(−2.53e4−4.38e4i)T2 |
| 41 | 1+396.T+6.89e4T2 |
| 43 | 1−260T+7.95e4T2 |
| 47 | 1+(−206.+357.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1+(238.+412.i)T+(−7.44e4+1.28e5i)T2 |
| 59 | 1+(142.+247.i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(−161+278.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(178+308.i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1−1.12e3T+3.57e5T2 |
| 73 | 1+(−113−195.i)T+(−1.94e5+3.36e5i)T2 |
| 79 | 1+(220−381.i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1−253.T+5.71e5T2 |
| 89 | 1+(−103.+178.i)T+(−3.52e5−6.10e5i)T2 |
| 97 | 1+1.33e3T+9.12e5T2 |
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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
−8.630888996559014389027070873089, −8.172982936258391272194249122369, −6.92211242074298463264052321937, −6.25228456213774353295466244111, −5.24169616775711685949650710337, −4.72119162388766358636475585407, −3.71711629805521054744976867424, −2.36102949601614990859665008211, −1.45509538617499389447480383563, −0.41879116089297839315015161529,
1.26192970783408412628407009884, 2.36978066036848677095724844974, 3.17843069620267052087326630378, 4.12456863638752191576269215668, 5.29524071945296012690776808714, 6.29694916414721993012213818685, 6.58271843551666450876949848066, 7.50230698055538417304384181744, 8.604676416704166700866973293836, 9.103462927790234235897909923075