L(s) = 1 | + (−0.753 − 0.657i)2-s + (0.134 − 0.990i)3-s + (0.134 + 0.990i)4-s + (−0.393 + 0.919i)5-s + (−0.753 + 0.657i)6-s + (0.691 − 0.722i)7-s + (0.550 − 0.834i)8-s + (−0.963 − 0.266i)9-s + (0.900 − 0.433i)10-s + 12-s + (0.0448 − 0.998i)13-s + (−0.995 + 0.0896i)14-s + (0.858 + 0.512i)15-s + (−0.963 + 0.266i)16-s + (0.809 + 0.587i)17-s + (0.550 + 0.834i)18-s + ⋯ |
L(s) = 1 | + (−0.753 − 0.657i)2-s + (0.134 − 0.990i)3-s + (0.134 + 0.990i)4-s + (−0.393 + 0.919i)5-s + (−0.753 + 0.657i)6-s + (0.691 − 0.722i)7-s + (0.550 − 0.834i)8-s + (−0.963 − 0.266i)9-s + (0.900 − 0.433i)10-s + 12-s + (0.0448 − 0.998i)13-s + (−0.995 + 0.0896i)14-s + (0.858 + 0.512i)15-s + (−0.963 + 0.266i)16-s + (0.809 + 0.587i)17-s + (0.550 + 0.834i)18-s + ⋯ |
Λ(s)=(=(319s/2ΓR(s+1)L(s)(−0.285−0.958i)Λ(1−s)
Λ(s)=(=(319s/2ΓR(s+1)L(s)(−0.285−0.958i)Λ(1−s)
Degree: |
1 |
Conductor: |
319
= 11⋅29
|
Sign: |
−0.285−0.958i
|
Analytic conductor: |
34.2813 |
Root analytic conductor: |
34.2813 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ319(226,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 319, (1: ), −0.285−0.958i)
|
Particular Values
L(21) |
≈ |
0.7808055129−1.047577238i |
L(21) |
≈ |
0.7808055129−1.047577238i |
L(1) |
≈ |
0.7126061889−0.4254585153i |
L(1) |
≈ |
0.7126061889−0.4254585153i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 29 | 1 |
good | 2 | 1+(−0.753−0.657i)T |
| 3 | 1+(0.134−0.990i)T |
| 5 | 1+(−0.393+0.919i)T |
| 7 | 1+(0.691−0.722i)T |
| 13 | 1+(0.0448−0.998i)T |
| 17 | 1+(0.809+0.587i)T |
| 19 | 1+(0.691+0.722i)T |
| 23 | 1+(−0.222+0.974i)T |
| 31 | 1+(0.753+0.657i)T |
| 37 | 1+(0.936−0.351i)T |
| 41 | 1+(−0.309−0.951i)T |
| 43 | 1+(0.222−0.974i)T |
| 47 | 1+(0.936+0.351i)T |
| 53 | 1+(0.753+0.657i)T |
| 59 | 1+(0.309−0.951i)T |
| 61 | 1+(−0.473−0.880i)T |
| 67 | 1+(0.623+0.781i)T |
| 71 | 1+(−0.963+0.266i)T |
| 73 | 1+(−0.858−0.512i)T |
| 79 | 1+(0.963+0.266i)T |
| 83 | 1+(−0.983−0.178i)T |
| 89 | 1+(−0.222−0.974i)T |
| 97 | 1+(0.473−0.880i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.19083239603087323064681282799, −24.478817319134056964709547139113, −23.68345196595970082513476884751, −22.62331421021131846291505445311, −21.40415854271471260611991851516, −20.66491226677822689519445344880, −19.87093341347340481860547020881, −18.84415065268353937923708949404, −17.863638870838617684731293260188, −16.69473826446705207215154812074, −16.32657398562973954706256499566, −15.37681371115101570843485848934, −14.64521776476250946707292015079, −13.67310580559823180962530897122, −11.849341141297377877873471568734, −11.3482789905891870954346575299, −9.942911835507119064338207789076, −9.16375391555208521039879373073, −8.50620254046276473933059498092, −7.62091125143771760486916282430, −6.027671264131208481811945813294, −5.0163056380491732050287308790, −4.40286439830944039882270776307, −2.51460156403157096584480824048, −0.949745589581379588668620942797,
0.650174316098318877372499513205, 1.698737527809078851715210944544, 2.98750335360999522215647347783, 3.8090074083564682560546623840, 5.77752232428223107924858157081, 7.21196846160991300383546162193, 7.67510444010936621875707585023, 8.42214758901098143744035029928, 10.011863606550015893045757809315, 10.7674448969482915099474009, 11.6831270438879119012641187916, 12.43609493322299366773704808068, 13.624111304768036258082211114166, 14.393516688888980098489163689768, 15.62914317300535593636218095382, 17.051843022450937842187279743348, 17.69705168995880148479052576976, 18.440162493409890415277792392699, 19.20715657290529257249738738859, 20.01159634102255636616191339539, 20.73216775712928688035785003275, 21.95904819915635022729682043151, 22.990131129327854612189178029501, 23.649120318951640887675036791631, 24.93616355279682023639640929963