L(s) = 1 | + (0.955 + 0.294i)3-s + (−0.733 + 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.955 + 0.294i)33-s + (0.988 + 0.149i)37-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)3-s + (−0.733 + 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.955 + 0.294i)33-s + (0.988 + 0.149i)37-s + ⋯ |
Λ(s)=(=(392s/2ΓR(s+1)L(s)(−0.0427−0.999i)Λ(1−s)
Λ(s)=(=(392s/2ΓR(s+1)L(s)(−0.0427−0.999i)Λ(1−s)
Degree: |
1 |
Conductor: |
392
= 23⋅72
|
Sign: |
−0.0427−0.999i
|
Analytic conductor: |
42.1262 |
Root analytic conductor: |
42.1262 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ392(285,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 392, (1: ), −0.0427−0.999i)
|
Particular Values
L(21) |
≈ |
0.6803427533−0.7100621476i |
L(21) |
≈ |
0.6803427533−0.7100621476i |
L(1) |
≈ |
1.032718105+0.07274365166i |
L(1) |
≈ |
1.032718105+0.07274365166i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+(0.955+0.294i)T |
| 5 | 1+(−0.733+0.680i)T |
| 11 | 1+(−0.826+0.563i)T |
| 13 | 1+(−0.900−0.433i)T |
| 17 | 1+(−0.365−0.930i)T |
| 19 | 1+(−0.5−0.866i)T |
| 23 | 1+(0.365−0.930i)T |
| 29 | 1+(−0.623+0.781i)T |
| 31 | 1+(0.5−0.866i)T |
| 37 | 1+(0.988+0.149i)T |
| 41 | 1+(0.222+0.974i)T |
| 43 | 1+(0.222−0.974i)T |
| 47 | 1+(−0.0747−0.997i)T |
| 53 | 1+(0.988−0.149i)T |
| 59 | 1+(−0.733−0.680i)T |
| 61 | 1+(−0.988−0.149i)T |
| 67 | 1+(0.5−0.866i)T |
| 71 | 1+(0.623+0.781i)T |
| 73 | 1+(−0.0747+0.997i)T |
| 79 | 1+(−0.5−0.866i)T |
| 83 | 1+(−0.900+0.433i)T |
| 89 | 1+(−0.826−0.563i)T |
| 97 | 1−T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.32569823829457606669070214255, −23.890294364217125712400107063208, −22.94852095625988070346539524684, −21.40449217972606616826388985054, −21.075196952865455299948155853918, −19.87274427596621910052396979072, −19.37417103956034881019573753925, −18.66424865356228416610773772457, −17.401236094232942525037645702500, −16.41798582445063492773953376982, −15.47596387217556112720457630143, −14.807740739952319075412642334805, −13.702624952224089191945761488125, −12.85107897866236963981929950114, −12.181633068402145709998433483921, −10.964108534260790000444265173919, −9.7568616391741779502001063654, −8.80690690652337987220427053568, −7.99803231294197944412544023971, −7.36298549520942760628970661329, −5.93799405308837629906364904198, −4.57888460502474408550866419964, −3.71407262955854739303469639222, −2.53197098480124441459900202335, −1.298583149487686324574490623322,
0.23470224434600038651792229127, 2.40576387064497229843928732189, 2.8665908927099713649924344920, 4.23318507138911775711811437962, 5.01092283770255809302903820637, 6.85799027297320002989798592919, 7.49831870292937420569849755207, 8.370744811124707058502618431331, 9.50659677976628763963701476157, 10.37276193718787323559756744202, 11.24592593901866821785960374003, 12.51977255790164820493661064315, 13.343556417018077204386998640638, 14.485167960505510344301308692678, 15.15160273099884249872663883000, 15.68487173351106757779538433587, 16.844269469076334241897774542835, 18.237455082151163479241012752769, 18.730057552480973434291833174888, 19.94949168627893550722555915113, 20.18120806189071127716387678678, 21.39309837176308404186420947904, 22.25533892978246645813890123136, 23.05417396729178609325645054616, 24.11696398059342280596788507131