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Properties

Label 1-163-163.92-r1-0-0
Degree $1$
Conductor $163$
Sign $0.985 + 0.170i$
Analytic cond. $17.5167$
Root an. cond. $17.5167$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

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: $\chi_{163} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (1:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.894396498 + 0.4190967117i\)
\(L(\frac12)\) \(\approx\) \(4.894396498 + 0.4190967117i\)
\(L(1)\) \(\approx\) \(2.684621363 + 0.1851349916i\)
\(L(1)\) \(\approx\) \(2.684621363 + 0.1851349916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.910 + 0.413i)T \)
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 \( 1 + (-0.740 - 0.672i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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

Graph of the $Z$-function along the critical line