L(s) = 1 | + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 − 0.866i)17-s + (−0.988 − 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)27-s + (0.0747 − 0.997i)31-s + (−0.988 − 0.149i)33-s + (−0.955 + 0.294i)37-s + (0.365 + 0.930i)39-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 − 0.866i)17-s + (−0.988 − 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)27-s + (0.0747 − 0.997i)31-s + (−0.988 − 0.149i)33-s + (−0.955 + 0.294i)37-s + (0.365 + 0.930i)39-s + ⋯ |
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.799−0.601i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.799−0.601i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
−0.799−0.601i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(51,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), −0.799−0.601i)
|
Particular Values
L(21) |
≈ |
0.1558354382−0.4664121535i |
L(21) |
≈ |
0.1558354382−0.4664121535i |
L(1) |
≈ |
0.8198278747+0.006356008739i |
L(1) |
≈ |
0.8198278747+0.006356008739i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
| 29 | 1 |
good | 3 | 1+(−0.988+0.149i)T |
| 5 | 1+(0.826+0.563i)T |
| 11 | 1+(0.955+0.294i)T |
| 13 | 1+(−0.222−0.974i)T |
| 17 | 1+(0.5−0.866i)T |
| 19 | 1+(−0.988−0.149i)T |
| 23 | 1+(−0.0747−0.997i)T |
| 31 | 1+(0.0747−0.997i)T |
| 37 | 1+(−0.955+0.294i)T |
| 41 | 1−T |
| 43 | 1+(−0.900−0.433i)T |
| 47 | 1+(−0.733+0.680i)T |
| 53 | 1+(0.0747−0.997i)T |
| 59 | 1+(0.5−0.866i)T |
| 61 | 1+(−0.365+0.930i)T |
| 67 | 1+(0.733+0.680i)T |
| 71 | 1+(0.222+0.974i)T |
| 73 | 1+(−0.826+0.563i)T |
| 79 | 1+(0.955−0.294i)T |
| 83 | 1+(−0.623+0.781i)T |
| 89 | 1+(−0.826−0.563i)T |
| 97 | 1+(−0.623+0.781i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.17003634714836981200357139040, −21.48014621167595615583934453864, −21.22997827501073372155176588340, −19.777998934676991018769763618624, −19.135204707662800558886634274723, −18.19077222452700048996399827904, −17.24637427330324944494767266127, −16.92686016084669263967618190163, −16.25929960036928622217465387601, −15.08599144558639533452532687721, −14.07576214730531707913935891311, −13.34446217145436382810318818178, −12.36480729692755199088784586108, −11.881759838475161010245552323486, −10.82646987501170492945985208513, −10.01964134706044673174417871590, −9.1856190649771899902612262791, −8.2838083419890127366036103012, −6.88459902872294734866747365662, −6.333961176762971478148541521568, −5.47735961924563113061741085629, −4.61548693231450455146435867545, −3.64327486201707117996173665720, −1.802272846948377334567210212025, −1.39235333905559225638131489973,
0.12386831344812600006931317190, 1.34019451265072274498330152508, 2.48874868856449713808025622299, 3.709185590903733997070110628711, 4.85026725113815546147308585804, 5.61847178968723910083117385503, 6.55893712107154699227859931734, 7.03112057744706380218394748006, 8.40195906656099230800261299733, 9.664513343979317211675749034471, 10.104503199942107532435539266829, 10.96968497020187775938022403890, 11.81100631782120833921036960480, 12.65315835292740419183603217750, 13.46488624227166270672254897233, 14.57807763902686406125786704716, 15.124576348470050106359188015881, 16.24890195815007706158326536, 17.12955994987080210534129105744, 17.492039065493688996378479276799, 18.40895628519290153512382567798, 19.030539232013679840938683140, 20.31149198542435615593778966465, 21.02236637884526625189396960534, 21.954955478252055628889348243120