L(s) = 1 | + (−0.474 + 0.930i)2-s + (−0.440 − 2.78i)3-s + (0.533 + 0.734i)4-s + (2.23 − 0.0540i)5-s + (2.79 + 0.909i)6-s + (−0.543 − 0.0860i)7-s + (−3.00 + 0.475i)8-s + (−4.68 + 1.52i)9-s + (−1.01 + 2.10i)10-s + (−3.29 + 0.335i)11-s + (1.80 − 1.80i)12-s + (2.89 + 1.47i)13-s + (0.337 − 0.464i)14-s + (−1.13 − 6.19i)15-s + (0.419 − 1.29i)16-s + (−5.04 + 2.57i)17-s + ⋯ |
L(s) = 1 | + (−0.335 + 0.658i)2-s + (−0.254 − 1.60i)3-s + (0.266 + 0.367i)4-s + (0.999 − 0.0241i)5-s + (1.14 + 0.371i)6-s + (−0.205 − 0.0325i)7-s + (−1.06 + 0.168i)8-s + (−1.56 + 0.507i)9-s + (−0.319 + 0.666i)10-s + (−0.994 + 0.101i)11-s + (0.522 − 0.522i)12-s + (0.801 + 0.408i)13-s + (0.0902 − 0.124i)14-s + (−0.293 − 1.59i)15-s + (0.104 − 0.322i)16-s + (−1.22 + 0.623i)17-s + ⋯ |
Λ(s)=(=(55s/2ΓC(s)L(s)(0.995+0.0917i)Λ(2−s)
Λ(s)=(=(55s/2ΓC(s+1/2)L(s)(0.995+0.0917i)Λ(1−s)
Degree: |
2 |
Conductor: |
55
= 5⋅11
|
Sign: |
0.995+0.0917i
|
Analytic conductor: |
0.439177 |
Root analytic conductor: |
0.662704 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ55(28,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 55, ( :1/2), 0.995+0.0917i)
|
Particular Values
L(1) |
≈ |
0.769382−0.0353856i |
L(21) |
≈ |
0.769382−0.0353856i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−2.23+0.0540i)T |
| 11 | 1+(3.29−0.335i)T |
good | 2 | 1+(0.474−0.930i)T+(−1.17−1.61i)T2 |
| 3 | 1+(0.440+2.78i)T+(−2.85+0.927i)T2 |
| 7 | 1+(0.543+0.0860i)T+(6.65+2.16i)T2 |
| 13 | 1+(−2.89−1.47i)T+(7.64+10.5i)T2 |
| 17 | 1+(5.04−2.57i)T+(9.99−13.7i)T2 |
| 19 | 1+(−1.25−0.914i)T+(5.87+18.0i)T2 |
| 23 | 1+(0.803+0.803i)T+23iT2 |
| 29 | 1+(−3.44+2.50i)T+(8.96−27.5i)T2 |
| 31 | 1+(0.509+1.56i)T+(−25.0+18.2i)T2 |
| 37 | 1+(0.149−0.945i)T+(−35.1−11.4i)T2 |
| 41 | 1+(−5.25+7.23i)T+(−12.6−38.9i)T2 |
| 43 | 1+(−2.55+2.55i)T−43iT2 |
| 47 | 1+(−4.02+0.636i)T+(44.6−14.5i)T2 |
| 53 | 1+(3.19−6.27i)T+(−31.1−42.8i)T2 |
| 59 | 1+(−3.97−5.47i)T+(−18.2+56.1i)T2 |
| 61 | 1+(8.75+2.84i)T+(49.3+35.8i)T2 |
| 67 | 1+(2.62−2.62i)T−67iT2 |
| 71 | 1+(−2.11+6.51i)T+(−57.4−41.7i)T2 |
| 73 | 1+(−1.57+9.96i)T+(−69.4−22.5i)T2 |
| 79 | 1+(−1.28−3.96i)T+(−63.9+46.4i)T2 |
| 83 | 1+(−5.14−10.0i)T+(−48.7+67.1i)T2 |
| 89 | 1−3.64iT−89T2 |
| 97 | 1+(−14.8−7.56i)T+(57.0+78.4i)T2 |
show more | |
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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
−15.51754999881598381413867315642, −13.87455343802352467968095923407, −13.11349212285347358121412436226, −12.23867769931468976673391616349, −10.83980658820952966517355792890, −8.923872544738054384216099954414, −7.80962763772320751283150377086, −6.64131331392991508178295589584, −5.91964563116307206628612997750, −2.30201715349521540754721535038,
2.90621412093884840572391895095, 5.03398526570061603130591842774, 6.18452953024848744760122570952, 8.926292763870742986135841980245, 9.776804341246773447516730564630, 10.61413970828910872431206934616, 11.25440439795185372817309568595, 13.06871289874405598854395185784, 14.43294072849954603034576975303, 15.68791991476255340378338810510