L(s) = 1 | + (−0.715 + 0.699i)2-s + (0.203 + 0.979i)3-s + (0.0227 − 0.999i)4-s + (0.538 − 0.842i)5-s + (−0.829 − 0.557i)6-s + (−0.419 + 0.907i)7-s + (0.682 + 0.730i)8-s + (−0.917 + 0.398i)9-s + (0.203 + 0.979i)10-s + (−0.917 + 0.398i)11-s + (0.983 − 0.181i)12-s + (0.113 + 0.993i)13-s + (−0.334 − 0.942i)14-s + (0.934 + 0.356i)15-s + (−0.998 − 0.0455i)16-s + (−0.775 − 0.631i)17-s + ⋯ |
L(s) = 1 | + (−0.715 + 0.699i)2-s + (0.203 + 0.979i)3-s + (0.0227 − 0.999i)4-s + (0.538 − 0.842i)5-s + (−0.829 − 0.557i)6-s + (−0.419 + 0.907i)7-s + (0.682 + 0.730i)8-s + (−0.917 + 0.398i)9-s + (0.203 + 0.979i)10-s + (−0.917 + 0.398i)11-s + (0.983 − 0.181i)12-s + (0.113 + 0.993i)13-s + (−0.334 − 0.942i)14-s + (0.934 + 0.356i)15-s + (−0.998 − 0.0455i)16-s + (−0.775 − 0.631i)17-s + ⋯ |
Λ(s)=(=(967s/2ΓR(s)L(s)(0.0778−0.996i)Λ(1−s)
Λ(s)=(=(967s/2ΓR(s)L(s)(0.0778−0.996i)Λ(1−s)
Degree: |
1 |
Conductor: |
967
|
Sign: |
0.0778−0.996i
|
Analytic conductor: |
4.49072 |
Root analytic conductor: |
4.49072 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ967(124,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 967, (0: ), 0.0778−0.996i)
|
Particular Values
L(21) |
≈ |
0.005076130489+0.004695394750i |
L(21) |
≈ |
0.005076130489+0.004695394750i |
L(1) |
≈ |
0.5017987590+0.3234067983i |
L(1) |
≈ |
0.5017987590+0.3234067983i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 967 | 1 |
good | 2 | 1+(−0.715+0.699i)T |
| 3 | 1+(0.203+0.979i)T |
| 5 | 1+(0.538−0.842i)T |
| 7 | 1+(−0.419+0.907i)T |
| 11 | 1+(−0.917+0.398i)T |
| 13 | 1+(0.113+0.993i)T |
| 17 | 1+(−0.775−0.631i)T |
| 19 | 1+(−0.949−0.313i)T |
| 23 | 1+(0.854+0.519i)T |
| 29 | 1+(−0.917−0.398i)T |
| 31 | 1+(0.291−0.956i)T |
| 37 | 1+(0.113+0.993i)T |
| 41 | 1+(−0.0682+0.997i)T |
| 43 | 1+(−0.158−0.987i)T |
| 47 | 1+(0.934+0.356i)T |
| 53 | 1+(−0.877+0.480i)T |
| 59 | 1+(−0.877−0.480i)T |
| 61 | 1+(0.898−0.439i)T |
| 67 | 1+(0.682−0.730i)T |
| 71 | 1+(0.962−0.269i)T |
| 73 | 1+(−0.877−0.480i)T |
| 79 | 1+(−0.877−0.480i)T |
| 83 | 1+(−0.829−0.557i)T |
| 89 | 1+(−0.974−0.225i)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
−20.9146493873674743119700932781, −20.22418670188656251099398956757, −19.38875329751521676066561624842, −18.86690462318033149513647641913, −18.10192037147538518892073332106, −17.47541108947784956173276433611, −16.89029015305250079718049973545, −15.69720469942264451112368490098, −14.59910545716281774718590521250, −13.66464086322519189607613874945, −12.90200916244594380318573246358, −12.69968919436226821324153434414, −11.00052775507035253750508944415, −10.84378684703007096690644185835, −10.01886341693674828150213333676, −8.843938690188628154270672933146, −8.10174760538535465621105763470, −7.22040177494318374396853687175, −6.65966900389625082322215171677, −5.60176450169719615625619663216, −3.89739577950497153617380161987, −2.99703151788074125547638403845, −2.37119070489931065309838446031, −1.271457943929868121662026067983, −0.00358384427823642650083120979,
1.90355794966134076512879204488, 2.63738520388977160188865989081, 4.38838910074792950929399625793, 4.99084137105812992077925542217, 5.78342540088300965236957418509, 6.630992340264829692008187631754, 7.94893999987751025773907022259, 8.77500487977346916353974192522, 9.34413152146191311394476273284, 9.76574317751710082392480523795, 10.87184766821081184323906524200, 11.67161077907927115739543889985, 13.04918488803923792802650751325, 13.67209671474062467297444742287, 14.792957059782457985207685123911, 15.55778369026418654087833118511, 15.86392297117448484265371836032, 16.94745119630241800125665087471, 17.21686806401370252909854664629, 18.45779582642050908490745513867, 19.026860030976263277679121650676, 20.09277593524072748693275322541, 20.70766382387962778195708079170, 21.47819319101658183616642368140, 22.24089810047444343543257752252