L(s) = 1 | + (0.988 + 1.01i)2-s + (1.88 + 0.612i)3-s + (−0.0464 + 1.99i)4-s + (0.809 − 0.587i)5-s + (1.24 + 2.51i)6-s + (−0.668 − 2.05i)7-s + (−2.06 + 1.92i)8-s + (0.755 + 0.548i)9-s + (1.39 + 0.237i)10-s + (−3.31 − 0.105i)11-s + (−1.31 + 3.74i)12-s + (1.85 − 2.55i)13-s + (1.42 − 2.70i)14-s + (1.88 − 0.612i)15-s + (−3.99 − 0.185i)16-s + (−2.62 − 3.60i)17-s + ⋯ |
L(s) = 1 | + (0.698 + 0.715i)2-s + (1.08 + 0.353i)3-s + (−0.0232 + 0.999i)4-s + (0.361 − 0.262i)5-s + (0.507 + 1.02i)6-s + (−0.252 − 0.777i)7-s + (−0.731 + 0.682i)8-s + (0.251 + 0.182i)9-s + (0.440 + 0.0750i)10-s + (−0.999 − 0.0316i)11-s + (−0.379 + 1.08i)12-s + (0.515 − 0.709i)13-s + (0.379 − 0.724i)14-s + (0.487 − 0.158i)15-s + (−0.998 − 0.0464i)16-s + (−0.635 − 0.874i)17-s + ⋯ |
Λ(s)=(=(220s/2ΓC(s)L(s)(0.424−0.905i)Λ(2−s)
Λ(s)=(=(220s/2ΓC(s+1/2)L(s)(0.424−0.905i)Λ(1−s)
Degree: |
2 |
Conductor: |
220
= 22⋅5⋅11
|
Sign: |
0.424−0.905i
|
Analytic conductor: |
1.75670 |
Root analytic conductor: |
1.32540 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ220(51,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 220, ( :1/2), 0.424−0.905i)
|
Particular Values
L(1) |
≈ |
1.85961+1.18171i |
L(21) |
≈ |
1.85961+1.18171i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.988−1.01i)T |
| 5 | 1+(−0.809+0.587i)T |
| 11 | 1+(3.31+0.105i)T |
good | 3 | 1+(−1.88−0.612i)T+(2.42+1.76i)T2 |
| 7 | 1+(0.668+2.05i)T+(−5.66+4.11i)T2 |
| 13 | 1+(−1.85+2.55i)T+(−4.01−12.3i)T2 |
| 17 | 1+(2.62+3.60i)T+(−5.25+16.1i)T2 |
| 19 | 1+(1.21−3.74i)T+(−15.3−11.1i)T2 |
| 23 | 1−8.67iT−23T2 |
| 29 | 1+(−6.12+1.99i)T+(23.4−17.0i)T2 |
| 31 | 1+(−0.673+0.927i)T+(−9.57−29.4i)T2 |
| 37 | 1+(−3.17−9.78i)T+(−29.9+21.7i)T2 |
| 41 | 1+(1.29+0.421i)T+(33.1+24.0i)T2 |
| 43 | 1−4.00T+43T2 |
| 47 | 1+(3.61+1.17i)T+(38.0+27.6i)T2 |
| 53 | 1+(−6.21−4.51i)T+(16.3+50.4i)T2 |
| 59 | 1+(0.112−0.0366i)T+(47.7−34.6i)T2 |
| 61 | 1+(7.07+9.74i)T+(−18.8+58.0i)T2 |
| 67 | 1+6.69iT−67T2 |
| 71 | 1+(−0.556−0.766i)T+(−21.9+67.5i)T2 |
| 73 | 1+(−1.45+0.471i)T+(59.0−42.9i)T2 |
| 79 | 1+(−3.49−2.53i)T+(24.4+75.1i)T2 |
| 83 | 1+(0.790−0.574i)T+(25.6−78.9i)T2 |
| 89 | 1+13.2T+89T2 |
| 97 | 1+(−11.7−8.52i)T+(29.9+92.2i)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
−13.01062197929410793136657862035, −11.69342239676581801482713315531, −10.32826045282182695817912270109, −9.350088775793844800735219219068, −8.246796641625561920017469927753, −7.63454784091303112640316007609, −6.25058836712704833631221876392, −5.04593010799797638107837799630, −3.77419265402767692236574947123, −2.77753992010886957018276342360,
2.21105160766167002279830923842, 2.81681413601261214230889203425, 4.39131578465239767492875041701, 5.82081666678347137614169709957, 6.84666500308664187067667663233, 8.518737626908185358750470127277, 9.051665984436130551010411916803, 10.36369572226521495638502375288, 11.11417348943161230452723141651, 12.47607351664276414130043553955