L(s) = 1 | + (0.910 + 0.413i)2-s + (0.627 − 0.778i)3-s + (0.657 + 0.753i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)6-s + (0.963 − 0.268i)7-s + (0.286 + 0.957i)8-s + (−0.211 − 0.977i)9-s + (0.856 + 0.516i)10-s + (0.360 + 0.932i)11-s + (0.999 − 0.0387i)12-s + (−0.973 + 0.230i)13-s + (0.987 + 0.154i)14-s + (0.713 − 0.700i)15-s + (−0.135 + 0.990i)16-s + (0.686 − 0.727i)17-s + ⋯ |
L(s) = 1 | + (0.910 + 0.413i)2-s + (0.627 − 0.778i)3-s + (0.657 + 0.753i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)6-s + (0.963 − 0.268i)7-s + (0.286 + 0.957i)8-s + (−0.211 − 0.977i)9-s + (0.856 + 0.516i)10-s + (0.360 + 0.932i)11-s + (0.999 − 0.0387i)12-s + (−0.973 + 0.230i)13-s + (0.987 + 0.154i)14-s + (0.713 − 0.700i)15-s + (−0.135 + 0.990i)16-s + (0.686 − 0.727i)17-s + ⋯ |
Λ(s)=(=(163s/2ΓR(s+1)L(s)(0.985+0.170i)Λ(1−s)
Λ(s)=(=(163s/2ΓR(s+1)L(s)(0.985+0.170i)Λ(1−s)
Degree: |
1 |
Conductor: |
163
|
Sign: |
0.985+0.170i
|
Analytic conductor: |
17.5167 |
Root analytic conductor: |
17.5167 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ163(92,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 163, (1: ), 0.985+0.170i)
|
Particular Values
L(21) |
≈ |
4.894396498+0.4190967117i |
L(21) |
≈ |
4.894396498+0.4190967117i |
L(1) |
≈ |
2.684621363+0.1851349916i |
L(1) |
≈ |
2.684621363+0.1851349916i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 163 | 1 |
good | 2 | 1+(0.910+0.413i)T |
| 3 | 1+(0.627−0.778i)T |
| 5 | 1+(0.993+0.116i)T |
| 7 | 1+(0.963−0.268i)T |
| 11 | 1+(0.360+0.932i)T |
| 13 | 1+(−0.973+0.230i)T |
| 17 | 1+(0.686−0.727i)T |
| 19 | 1+(−0.996+0.0774i)T |
| 23 | 1+(−0.766−0.642i)T |
| 29 | 1+(−0.0968+0.995i)T |
| 31 | 1+(−0.396−0.918i)T |
| 37 | 1+(0.0581+0.998i)T |
| 41 | 1+(0.657−0.753i)T |
| 43 | 1+(−0.875−0.483i)T |
| 47 | 1+(0.813−0.581i)T |
| 53 | 1+(0.173−0.984i)T |
| 59 | 1+(0.5+0.866i)T |
| 61 | 1+(−0.286+0.957i)T |
| 67 | 1+(−0.323−0.946i)T |
| 71 | 1+(0.0193+0.999i)T |
| 73 | 1+(−0.323+0.946i)T |
| 79 | 1+(−0.952+0.305i)T |
| 83 | 1+(0.249−0.968i)T |
| 89 | 1+(0.360−0.932i)T |
| 97 | 1+(−0.740−0.672i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−27.66281546088581320827015426619, −26.51135781707470438583799448692, −25.18492055643018994390020096705, −24.71798125488778099324080450802, −23.60161307926149767786571933116, −22.03462444570796119673970702101, −21.57035125662190442477102025255, −21.0453754772945884018589183442, −19.90060982141481364098498723828, −19.0122445674196507215874319042, −17.42611732666492418300963480488, −16.382109561357674892280204399636, −15.02099271590655411307797315919, −14.41735531307991376418590193626, −13.67714273145879553463712820892, −12.446171653036719463953988922765, −11.10878611807429393259896726569, −10.256214984324640467844035248789, −9.21518844294741713219246025052, −7.91523013533693751686301814516, −6.02074363986313919425215771651, −5.18044319099269481424227254108, −4.06424402044278021819650785294, −2.69232255206585238823983257286, −1.669875599721562727290466957995,
1.7479577122349737697322461860, 2.524890063729229419520600515, 4.238298210513452571969770202495, 5.43972948243493812043665506467, 6.76888062589188383125804190735, 7.47466108167697787765251967008, 8.72281073596334299210621659721, 10.146420194492285051373079976079, 11.77655123883400004767591266896, 12.599062379368713835822663226, 13.65479758349832742295906094182, 14.54737003464056227223968679265, 14.841981861933836545870956362275, 16.80670022321365896016448195340, 17.503886282306088687585033531621, 18.48739091187860763097634842790, 20.09587378971810506152900303094, 20.69821710295946624924722083314, 21.67960278038063505647936229219, 22.76106475270274997931682790125, 23.92053228785366930836573786200, 24.53205433123412010777031835146, 25.45106454154881525921017998161, 25.98968039107767569050851320543, 27.330045750447379061753021837198