L(s) = 1 | + (−0.997 + 0.0712i)2-s + (0.878 + 0.478i)3-s + (0.989 − 0.142i)4-s + (0.0630 + 0.998i)5-s + (−0.909 − 0.414i)6-s + (−0.977 + 0.212i)8-s + (0.542 + 0.840i)9-s + (−0.133 − 0.990i)10-s + (−0.907 − 0.419i)11-s + (0.937 + 0.348i)12-s + (0.692 − 0.721i)13-s + (−0.422 + 0.906i)15-s + (0.959 − 0.281i)16-s + (−0.112 + 0.993i)17-s + (−0.600 − 0.799i)18-s + (−0.336 − 0.941i)19-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0712i)2-s + (0.878 + 0.478i)3-s + (0.989 − 0.142i)4-s + (0.0630 + 0.998i)5-s + (−0.909 − 0.414i)6-s + (−0.977 + 0.212i)8-s + (0.542 + 0.840i)9-s + (−0.133 − 0.990i)10-s + (−0.907 − 0.419i)11-s + (0.937 + 0.348i)12-s + (0.692 − 0.721i)13-s + (−0.422 + 0.906i)15-s + (0.959 − 0.281i)16-s + (−0.112 + 0.993i)17-s + (−0.600 − 0.799i)18-s + (−0.336 − 0.941i)19-s + ⋯ |
Λ(s)=(=(2681s/2ΓR(s+1)L(s)(0.00926−0.999i)Λ(1−s)
Λ(s)=(=(2681s/2ΓR(s+1)L(s)(0.00926−0.999i)Λ(1−s)
Degree: |
1 |
Conductor: |
2681
= 7⋅383
|
Sign: |
0.00926−0.999i
|
Analytic conductor: |
288.113 |
Root analytic conductor: |
288.113 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2681(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 2681, (1: ), 0.00926−0.999i)
|
Particular Values
L(21) |
≈ |
0.2098831068−0.2079469585i |
L(21) |
≈ |
0.2098831068−0.2079469585i |
L(1) |
≈ |
0.7713022288+0.2696580901i |
L(1) |
≈ |
0.7713022288+0.2696580901i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 383 | 1 |
good | 2 | 1+(−0.997+0.0712i)T |
| 3 | 1+(0.878+0.478i)T |
| 5 | 1+(0.0630+0.998i)T |
| 11 | 1+(−0.907−0.419i)T |
| 13 | 1+(0.692−0.721i)T |
| 17 | 1+(−0.112+0.993i)T |
| 19 | 1+(−0.336−0.941i)T |
| 23 | 1+(−0.320−0.947i)T |
| 29 | 1+(−0.330−0.943i)T |
| 31 | 1+(0.432+0.901i)T |
| 37 | 1+(0.622+0.782i)T |
| 41 | 1+(−0.551+0.834i)T |
| 43 | 1+(0.578+0.815i)T |
| 47 | 1+(0.225+0.974i)T |
| 53 | 1+(−0.246−0.969i)T |
| 59 | 1+(−0.971+0.238i)T |
| 61 | 1+(−0.978−0.206i)T |
| 67 | 1+(0.273−0.961i)T |
| 71 | 1+(0.864+0.502i)T |
| 73 | 1+(0.893+0.449i)T |
| 79 | 1+(−0.994+0.103i)T |
| 83 | 1+(−0.973−0.228i)T |
| 89 | 1+(−0.0684+0.997i)T |
| 97 | 1+(0.902−0.429i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−19.226219896505383760469419659912, −18.48133073346174681238399015627, −18.2246448790061337939199926171, −17.22059433478942262872733485631, −16.57446438091030984861181161649, −15.692972096680569921012942664485, −15.46379095456208218656296903306, −14.248787240940052646236648037975, −13.58255918124883291195158938926, −12.78946568838923124973114041973, −12.18804651193500876544150243607, −11.48106927276074158819441461717, −10.42725122187308689666349458559, −9.643429741304241798373537903443, −9.07027531956292462828518117397, −8.54280304360142856903934090832, −7.66288211437931823729397446690, −7.34927436233576112377981262259, −6.23266772512197685662447781923, −5.44374737608665035418127355739, −4.21899665305589687732219860030, −3.41650508883393631973879815893, −2.29468183372224260487173578425, −1.783593250581947111305642552273, −0.92898443577758410428396553195,
0.06063680028097741758625882016, 1.375793598362566159085237253462, 2.48518343781343143839870115706, 2.84578553984846670204198212069, 3.66038705323702303824124990993, 4.80853989661930143225413155681, 6.03752854632176316799271099410, 6.51257265083039196070225123846, 7.62601145153112486394419084535, 8.12067721189041442361508167967, 8.632427684255914561335053844018, 9.66811214605900118664306171242, 10.23279524998153048142949231482, 10.875335653480797102675695737, 11.20683194606278665737194961288, 12.62037381668679624112908885897, 13.34008350439679917368875931887, 14.186361007993545288463713725876, 14.97479721832676894165307253026, 15.48613079885268130568827805609, 15.901868874120861033498517978258, 16.891857393828174429426892652221, 17.68269380101156738227094324810, 18.48731606955741393820516600945, 18.77618800777870709707155653977