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Properties

Label 1-163-163.113-r0-0-0
Degree $1$
Conductor $163$
Sign $0.175 + 0.984i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (−0.360 − 0.932i)2-s + (0.466 − 0.884i)3-s + (−0.740 + 0.672i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (−0.981 + 0.192i)7-s + (0.893 + 0.448i)8-s + (−0.565 − 0.824i)9-s + (0.925 + 0.378i)10-s + (−0.999 + 0.0387i)11-s + (0.249 + 0.968i)12-s + (−0.0581 + 0.998i)13-s + (0.533 + 0.845i)14-s + (0.323 + 0.946i)15-s + (0.0968 − 0.995i)16-s + (−0.835 + 0.549i)17-s + ⋯
L(s)  = 1  + (−0.360 − 0.932i)2-s + (0.466 − 0.884i)3-s + (−0.740 + 0.672i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (−0.981 + 0.192i)7-s + (0.893 + 0.448i)8-s + (−0.565 − 0.824i)9-s + (0.925 + 0.378i)10-s + (−0.999 + 0.0387i)11-s + (0.249 + 0.968i)12-s + (−0.0581 + 0.998i)13-s + (0.533 + 0.845i)14-s + (0.323 + 0.946i)15-s + (0.0968 − 0.995i)16-s + (−0.835 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1360035615 + 0.1138516879i\)
\(L(\frac12)\) \(\approx\) \(0.1360035615 + 0.1138516879i\)
\(L(1)\) \(\approx\) \(0.4942275253 - 0.2032195472i\)
\(L(1)\) \(\approx\) \(0.4942275253 - 0.2032195472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.360 - 0.932i)T \)
3 \( 1 + (0.466 - 0.884i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (-0.981 + 0.192i)T \)
11 \( 1 + (-0.999 + 0.0387i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.835 + 0.549i)T \)
19 \( 1 + (-0.875 + 0.483i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.987 - 0.154i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
37 \( 1 + (0.396 - 0.918i)T \)
41 \( 1 + (-0.740 - 0.672i)T \)
43 \( 1 + (-0.135 + 0.990i)T \)
47 \( 1 + (-0.627 + 0.778i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (-0.211 + 0.977i)T \)
71 \( 1 + (-0.790 - 0.612i)T \)
73 \( 1 + (-0.211 - 0.977i)T \)
79 \( 1 + (-0.431 - 0.902i)T \)
83 \( 1 + (0.657 + 0.753i)T \)
89 \( 1 + (-0.999 - 0.0387i)T \)
97 \( 1 + (0.996 + 0.0774i)T \)
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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

−27.16661056908369371362259051060, −26.66603499901268174347022258145, −25.60300917469134838062391015956, −24.92332880271639423695899913398, −23.62426579877138283777811023828, −22.886772899064648190180787502990, −21.858262734590506784047580681578, −20.34448941236308847724398469264, −19.79491933399956192566298608081, −18.70152026724253983470621278627, −17.28774936385313750508272216263, −16.27003288778411314043892693201, −15.72129423837857986383964612197, −15.03786795008434483609069255197, −13.54306652062709273813620392816, −12.78100604062842541787930027270, −10.77476741719162897154200237634, −9.95233291761905781243284460072, −8.756102007867448100400801945919, −8.158315905163692996670102603, −6.794215897790735277107539467391, −5.23084290049047091691434624322, −4.44326917637878683133967960001, −3.00052829587396510435517766026, −0.14808551844264482361106118752, 2.05551312537592872351229243455, 2.99788815535701475719580887417, 4.09271134411277814111489399863, 6.308883219301649725154902176038, 7.42936515698726515955344614724, 8.41316874212264044413735598814, 9.53778776394343854419677312694, 10.78029230651997046029565230490, 11.81129110067092408875040271013, 12.78338293038697919701326331253, 13.53637643195723676755062456699, 14.80333936118807907788918050943, 16.07887521455071920940741197216, 17.53398560129597104494711919059, 18.532024358105963817385458769224, 19.23469692651270204850677350201, 19.649070714647773689762545052330, 20.96509075343345503662685768744, 22.035833942921137549269011950870, 23.16110426760550354774498883331, 23.7501380516356694225212415129, 25.47766317141090123487702277919, 26.15055327288322924732145562332, 26.78642559898843646195040449143, 28.21576988952718675800304864572

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