L(s) = 1 | + (1.03 − 1.23i)2-s + (1.23 − 1.21i)3-s + (−0.105 − 0.597i)4-s + (−0.212 − 2.78i)6-s + (1.71 + 0.302i)7-s + (1.94 + 1.12i)8-s + (0.0676 − 2.99i)9-s + (2.93 − 1.06i)11-s + (−0.853 − 0.612i)12-s + (1.21 + 1.44i)13-s + (2.15 − 1.80i)14-s + (4.55 − 1.65i)16-s + (−5.56 + 3.21i)17-s + (−3.63 − 3.19i)18-s + (−2.43 + 4.21i)19-s + ⋯ |
L(s) = 1 | + (0.733 − 0.874i)2-s + (0.715 − 0.699i)3-s + (−0.0526 − 0.298i)4-s + (−0.0866 − 1.13i)6-s + (0.647 + 0.114i)7-s + (0.688 + 0.397i)8-s + (0.0225 − 0.999i)9-s + (0.884 − 0.321i)11-s + (−0.246 − 0.176i)12-s + (0.337 + 0.402i)13-s + (0.575 − 0.482i)14-s + (1.13 − 0.414i)16-s + (−1.34 + 0.778i)17-s + (−0.857 − 0.753i)18-s + (−0.558 + 0.967i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(0.109+0.993i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(0.109+0.993i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
0.109+0.993i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(124,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), 0.109+0.993i)
|
Particular Values
L(1) |
≈ |
2.29922−2.05963i |
L(21) |
≈ |
2.29922−2.05963i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−1.23+1.21i)T |
| 5 | 1 |
good | 2 | 1+(−1.03+1.23i)T+(−0.347−1.96i)T2 |
| 7 | 1+(−1.71−0.302i)T+(6.57+2.39i)T2 |
| 11 | 1+(−2.93+1.06i)T+(8.42−7.07i)T2 |
| 13 | 1+(−1.21−1.44i)T+(−2.25+12.8i)T2 |
| 17 | 1+(5.56−3.21i)T+(8.5−14.7i)T2 |
| 19 | 1+(2.43−4.21i)T+(−9.5−16.4i)T2 |
| 23 | 1+(2.73−0.481i)T+(21.6−7.86i)T2 |
| 29 | 1+(5.94+4.98i)T+(5.03+28.5i)T2 |
| 31 | 1+(0.687+3.89i)T+(−29.1+10.6i)T2 |
| 37 | 1+(0.704−0.406i)T+(18.5−32.0i)T2 |
| 41 | 1+(−5.80+4.86i)T+(7.11−40.3i)T2 |
| 43 | 1+(0.901+2.47i)T+(−32.9+27.6i)T2 |
| 47 | 1+(7.03+1.24i)T+(44.1+16.0i)T2 |
| 53 | 1+1.86iT−53T2 |
| 59 | 1+(0.971+0.353i)T+(45.1+37.9i)T2 |
| 61 | 1+(1.78−10.1i)T+(−57.3−20.8i)T2 |
| 67 | 1+(−8.76−10.4i)T+(−11.6+65.9i)T2 |
| 71 | 1+(−4.06−7.03i)T+(−35.5+61.4i)T2 |
| 73 | 1+(8.53+4.92i)T+(36.5+63.2i)T2 |
| 79 | 1+(−11.3−9.56i)T+(13.7+77.7i)T2 |
| 83 | 1+(5.92−7.06i)T+(−14.4−81.7i)T2 |
| 89 | 1+(−6.28+10.8i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−5.39−14.8i)T+(−74.3+62.3i)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
−10.59222256604962832598761198935, −9.381235023070166687969730680124, −8.460290181229207125376246120096, −7.88352272667074443441799877890, −6.70493648518675406553455365033, −5.77087312630096008690946615762, −4.12236282268814924399541828136, −3.83438007716546518105027965209, −2.27634193278807308696379066844, −1.64237742594467920450898533647,
1.87198864730275629675092479639, 3.45509220535545195524857332688, 4.57395793662979583730055541035, 4.89882560258140593896985780450, 6.24995230709484139308407568120, 7.09682729025215052426545002785, 7.980759936550222676419818468592, 8.939723581205023805014841435824, 9.636686647195102370198443578893, 10.84187428811258693405858759331