L(s) = 1 | + (1.36 + 1.36i)3-s + (4.24 + 2.64i)5-s + (4.16 − 4.16i)7-s − 5.29i·9-s + 3.31·11-s + (7.53 + 7.53i)13-s + (2.16 + 9.37i)15-s + (−10.2 + 10.2i)17-s + 6.46i·19-s + 11.3·21-s + (−0.810 − 0.810i)23-s + (10.9 + 22.4i)25-s + (19.4 − 19.4i)27-s − 2.81i·29-s + 13.1·31-s + ⋯ |
L(s) = 1 | + (0.453 + 0.453i)3-s + (0.848 + 0.529i)5-s + (0.594 − 0.594i)7-s − 0.588i·9-s + 0.301·11-s + (0.579 + 0.579i)13-s + (0.144 + 0.625i)15-s + (−0.605 + 0.605i)17-s + 0.340i·19-s + 0.539·21-s + (−0.0352 − 0.0352i)23-s + (0.439 + 0.898i)25-s + (0.720 − 0.720i)27-s − 0.0969i·29-s + 0.424·31-s + ⋯ |
Λ(s)=(=(220s/2ΓC(s)L(s)(0.896−0.443i)Λ(3−s)
Λ(s)=(=(220s/2ΓC(s+1)L(s)(0.896−0.443i)Λ(1−s)
Degree: |
2 |
Conductor: |
220
= 22⋅5⋅11
|
Sign: |
0.896−0.443i
|
Analytic conductor: |
5.99456 |
Root analytic conductor: |
2.44838 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ220(133,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 220, ( :1), 0.896−0.443i)
|
Particular Values
L(23) |
≈ |
2.11360+0.493956i |
L(21) |
≈ |
2.11360+0.493956i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(−4.24−2.64i)T |
| 11 | 1−3.31T |
good | 3 | 1+(−1.36−1.36i)T+9iT2 |
| 7 | 1+(−4.16+4.16i)T−49iT2 |
| 13 | 1+(−7.53−7.53i)T+169iT2 |
| 17 | 1+(10.2−10.2i)T−289iT2 |
| 19 | 1−6.46iT−361T2 |
| 23 | 1+(0.810+0.810i)T+529iT2 |
| 29 | 1+2.81iT−841T2 |
| 31 | 1−13.1T+961T2 |
| 37 | 1+(−4.46+4.46i)T−1.36e3iT2 |
| 41 | 1+71.2T+1.68e3T2 |
| 43 | 1+(17.9+17.9i)T+1.84e3iT2 |
| 47 | 1+(−9.43+9.43i)T−2.20e3iT2 |
| 53 | 1+(−1.65−1.65i)T+2.80e3iT2 |
| 59 | 1+35.5iT−3.48e3T2 |
| 61 | 1+71.8T+3.72e3T2 |
| 67 | 1+(−64.1+64.1i)T−4.48e3iT2 |
| 71 | 1+77.7T+5.04e3T2 |
| 73 | 1+(66.9+66.9i)T+5.32e3iT2 |
| 79 | 1+42.5iT−6.24e3T2 |
| 83 | 1+(58.8+58.8i)T+6.88e3iT2 |
| 89 | 1−111.iT−7.92e3T2 |
| 97 | 1+(−52.0+52.0i)T−9.40e3iT2 |
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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
−12.03096713513333828552122373817, −10.96172295408158773506846073458, −10.18000420095989614889444589553, −9.246885380984771035009134053465, −8.362636117114452423973286268437, −6.92114774608541240208634202296, −6.07309038589560335904590209274, −4.49083520960824496796110861056, −3.38234638682046916073462839153, −1.69541760760645339835147635779,
1.52140200820051163532608095565, 2.70981712925565931253065485483, 4.72692274548705372309127150574, 5.66371584154087404615200265276, 6.93699809445567701137687126172, 8.285104464577923062972439351834, 8.779683725577935127124302569220, 9.950209789123311984620734654781, 11.06590080723307797294134684875, 12.08661754246806697987948425179