L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + ⋯ |
Λ(s)=(=(23s/2ΓR(s)L(s)(0.986−0.165i)Λ(1−s)
Λ(s)=(=(23s/2ΓR(s)L(s)(0.986−0.165i)Λ(1−s)
Degree: |
1 |
Conductor: |
23
|
Sign: |
0.986−0.165i
|
Analytic conductor: |
0.106811 |
Root analytic conductor: |
0.106811 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ23(9,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 23, (0: ), 0.986−0.165i)
|
Particular Values
L(21) |
≈ |
0.8564154349−0.07119029211i |
L(21) |
≈ |
0.8564154349−0.07119029211i |
L(1) |
≈ |
1.130623330−0.08441276772i |
L(1) |
≈ |
1.130623330−0.08441276772i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 23 | 1 |
good | 2 | 1+(0.841−0.540i)T |
| 3 | 1+(−0.142+0.989i)T |
| 5 | 1+(−0.959+0.281i)T |
| 7 | 1+(−0.654−0.755i)T |
| 11 | 1+(0.841+0.540i)T |
| 13 | 1+(−0.654+0.755i)T |
| 17 | 1+(0.415+0.909i)T |
| 19 | 1+(0.415−0.909i)T |
| 29 | 1+(0.415+0.909i)T |
| 31 | 1+(−0.142−0.989i)T |
| 37 | 1+(−0.959−0.281i)T |
| 41 | 1+(−0.959+0.281i)T |
| 43 | 1+(−0.142+0.989i)T |
| 47 | 1+T |
| 53 | 1+(−0.654−0.755i)T |
| 59 | 1+(−0.654+0.755i)T |
| 61 | 1+(−0.142−0.989i)T |
| 67 | 1+(0.841−0.540i)T |
| 71 | 1+(0.841−0.540i)T |
| 73 | 1+(0.415−0.909i)T |
| 79 | 1+(−0.654+0.755i)T |
| 83 | 1+(−0.959−0.281i)T |
| 89 | 1+(−0.142+0.989i)T |
| 97 | 1+(−0.959+0.281i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−39.53057000057408974443827257399, −38.06405493719890636932453680584, −35.90801217877348028996675776859, −35.07041491708066181269011239110, −34.28831801930400569837137837857, −32.23632742777255278181244748430, −31.47207768496525831651942006288, −30.230316345376084729879785663673, −29.02871024721695166349870007671, −27.16701403565190250789603003861, −25.20173328033095296890873391070, −24.56382185594355984199241057903, −23.14452153449007311296374474522, −22.281227628668897155281978790810, −20.1602908688378450602602780520, −18.8444819164539367817621309389, −16.97064077543365217254061712072, −15.65507743925292049310893565467, −14.074306084298957911601624029266, −12.43425355313810107408547889512, −11.831741325069531222304676562549, −8.45117785217496318247771938581, −7.07465734175107573187136964239, −5.51944896175773567927746275915, −3.180067760838514227825587366893,
3.461093698942710071367139511604, 4.53726027325961264147771829576, 6.7760396547694166619140979147, 9.61449456153161740366302226097, 10.96302855097207220819127941529, 12.22204862691147626677748261347, 14.26452315674057170625267718594, 15.38415553822810622133363281295, 16.73628140793976823226951788309, 19.42429341204197976016823986381, 20.16997259280709505760435951272, 21.85239216769367598169346995665, 22.76816376270039974292210669320, 23.85940471327194475732402853122, 26.07174879497653635390264330688, 27.41593104953179932536090368156, 28.565267481620868950925252494262, 30.08605009406770738424560623997, 31.368129940671556856465118950469, 32.49414839088198040884830167161, 33.4264123054861695471883194583, 34.90206818040353173284656919011, 36.81912418317625834806522453641, 38.526479490849687529820230185270, 38.80237792185090640186982278813