L(s) = 1 | + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯ |
L(s) = 1 | + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯ |
Λ(s)=(=(405s/2ΓR(s+1)L(s)(−0.490+0.871i)Λ(1−s)
Λ(s)=(=(405s/2ΓR(s+1)L(s)(−0.490+0.871i)Λ(1−s)
Degree: |
1 |
Conductor: |
405
= 34⋅5
|
Sign: |
−0.490+0.871i
|
Analytic conductor: |
43.5232 |
Root analytic conductor: |
43.5232 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ405(148,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 405, (1: ), −0.490+0.871i)
|
Particular Values
L(21) |
≈ |
−0.06930522245−0.1185865562i |
L(21) |
≈ |
−0.06930522245−0.1185865562i |
L(1) |
≈ |
0.5359806133−0.2243017398i |
L(1) |
≈ |
0.5359806133−0.2243017398i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(−0.918−0.396i)T |
| 7 | 1+(0.230−0.973i)T |
| 11 | 1+(−0.835−0.549i)T |
| 13 | 1+(0.116−0.993i)T |
| 17 | 1+(−0.642−0.766i)T |
| 19 | 1+(−0.766−0.642i)T |
| 23 | 1+(0.230+0.973i)T |
| 29 | 1+(−0.597−0.802i)T |
| 31 | 1+(−0.286+0.957i)T |
| 37 | 1+(0.984−0.173i)T |
| 41 | 1+(0.396+0.918i)T |
| 43 | 1+(0.448+0.893i)T |
| 47 | 1+(−0.957+0.286i)T |
| 53 | 1+(0.866−0.5i)T |
| 59 | 1+(0.835−0.549i)T |
| 61 | 1+(−0.686+0.727i)T |
| 67 | 1+(−0.802−0.597i)T |
| 71 | 1+(−0.939−0.342i)T |
| 73 | 1+(−0.342−0.939i)T |
| 79 | 1+(−0.396+0.918i)T |
| 83 | 1+(0.918+0.396i)T |
| 89 | 1+(0.939−0.342i)T |
| 97 | 1+(0.998+0.0581i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.734830762352884407790830771151, −24.06351209975682946120103581113, −23.26666730334326776094572525988, −22.016153117910800065626966371256, −21.05228239507996475639322470333, −20.33866395498767985026021497758, −19.10204692259449993758505171614, −18.61948310371140654024392855156, −17.82451129683088305592071930104, −16.832862019712032792786530616882, −16.04015569045251653751388383236, −15.03352189102122191223591332747, −14.61059662574857998117779404046, −13.06814618993929078699251014259, −12.07936995436472836587430917942, −11.06031856618238038802857491192, −10.25194230076791623790352152382, −9.09561481829097412774649383698, −8.52096230977486058545475487877, −7.47720177346305807622267398758, −6.40715254235213466963011272394, −5.56205761193936535053268910156, −4.34474655301293708461074841341, −2.45322245845773125411715433728, −1.79111697659162914630577337601,
0.05596108957956790466864693234, 1.03136206126137478789981356739, 2.49611072505088546353564100728, 3.46488694007325160437877132443, 4.775348640252864484865426174432, 6.20843332856469335694456019570, 7.40097030656535824615365113564, 7.97497459612644936265762833585, 9.06692451593704286956053449427, 10.09942836853401088920666462997, 10.91376820002951390687602455310, 11.46538238839206640932826446687, 13.01713502892237984043956695031, 13.36712674152527561576033092587, 14.91877410541089758121604986252, 15.88588061871991013082028452025, 16.62788835885167413309963579106, 17.73385844121430591472462234957, 18.041102695151963679803583813548, 19.37362876835298105000022838967, 19.90902276844050445890780135435, 20.851112675470122767373845323508, 21.45141170146603046205072872120, 22.66440396088730810208619234346, 23.62235108137563508782640360764