L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.173 + 0.984i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.173 + 0.984i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯ |
Λ(s)=(=(740s/2ΓR(s)L(s)(0.379+0.925i)Λ(1−s)
Λ(s)=(=(740s/2ΓR(s)L(s)(0.379+0.925i)Λ(1−s)
Degree: |
1 |
Conductor: |
740
= 22⋅5⋅37
|
Sign: |
0.379+0.925i
|
Analytic conductor: |
3.43654 |
Root analytic conductor: |
3.43654 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ740(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 740, (0: ), 0.379+0.925i)
|
Particular Values
L(21) |
≈ |
0.3805085779+0.2551512424i |
L(21) |
≈ |
0.3805085779+0.2551512424i |
L(1) |
≈ |
0.6237727290−0.06450112604i |
L(1) |
≈ |
0.6237727290−0.06450112604i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 37 | 1 |
good | 3 | 1+(−0.766−0.642i)T |
| 7 | 1+(−0.939+0.342i)T |
| 11 | 1+(−0.5−0.866i)T |
| 13 | 1+(−0.984−0.173i)T |
| 17 | 1+(0.984−0.173i)T |
| 19 | 1+(0.642−0.766i)T |
| 23 | 1+(−0.866−0.5i)T |
| 29 | 1+(−0.866+0.5i)T |
| 31 | 1−iT |
| 41 | 1+(−0.173+0.984i)T |
| 43 | 1−iT |
| 47 | 1+(−0.5+0.866i)T |
| 53 | 1+(0.939+0.342i)T |
| 59 | 1+(−0.342+0.939i)T |
| 61 | 1+(−0.984−0.173i)T |
| 67 | 1+(0.939−0.342i)T |
| 71 | 1+(−0.766−0.642i)T |
| 73 | 1+T |
| 79 | 1+(0.342+0.939i)T |
| 83 | 1+(0.173+0.984i)T |
| 89 | 1+(−0.342+0.939i)T |
| 97 | 1+(0.866+0.5i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.431317280489186878310387839609, −21.690273793127680961101460426384, −20.69097096042691471794012372887, −20.128610269249596207212980811434, −19.0156041491345923344941456930, −18.25530760494049864998514120479, −17.169292275807635574382286988823, −16.759118752601551011930158923357, −15.86269393586173433683670315891, −15.17002902434725939189682344517, −14.25452047105888246504077622755, −13.10066906355784027131401684910, −12.24224004198375860436965808447, −11.73082333635335795572234197493, −10.33049640857923280593731313238, −9.97541459373026274066919633488, −9.33683798412951477765202763634, −7.73885895877651458684152668909, −7.0630322694922281235963998684, −5.9068222897472085104141775634, −5.26868311595961143633893284082, −4.1183201696772976561133363416, −3.39899866447145029245315165193, −1.98492192129756423706793500079, −0.2832173096051617890264876256,
1.00447296564445147406843746590, 2.48917086768761386895845018, 3.2745747912412164502744691366, 4.86682604450030438593166880797, 5.60832177286475627316693534851, 6.40020481671372623644693065380, 7.33211927160417182567912637227, 8.11043204629873593362099605573, 9.364627008314018199238890330891, 10.19122768660285331999383515142, 11.093616671372347087311085711801, 12.05860144021142526717099152718, 12.60981302149399372312041168838, 13.435621895830377388562948613023, 14.27692598593728889961806789873, 15.49105100375400413393815186736, 16.424291902620931120094662230415, 16.70567336904025233993484026004, 18.02765290170518006953792164524, 18.427701233841802589299241452354, 19.40509896194390664315199792374, 19.87419069142193243012419025453, 21.301226324529512713117025276040, 22.00080209640723417693769411597, 22.60835515930219675037145190286