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Properties

Label 1-163-163.71-r0-0-0
Degree $1$
Conductor $163$
Sign $0.859 - 0.511i$
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  + (−0.910 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3739132067 - 0.1028714510i\)
\(L(\frac12)\) \(\approx\) \(0.3739132067 - 0.1028714510i\)
\(L(1)\) \(\approx\) \(0.4600584273 + 0.01989031722i\)
\(L(1)\) \(\approx\) \(0.4600584273 + 0.01989031722i\)

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

−27.9664390596974796038601644584, −26.8124911991964121253640694455, −25.93243977583418955803306041730, −24.97340909682352443478761594035, −23.95871048139654755726092253226, −23.09054538850136351291766326384, −22.599607091064247574492123743905, −20.40187151345942364865092055447, −19.78541580870867932713642820282, −18.61166941594843274038149658987, −18.26396491504699437442816836541, −16.87317980833835447366018445811, −16.10373576759299486282039078228, −15.32020501192365229016815764389, −13.72656696110581938905776425031, −12.50496917335003295053057536123, −11.44930569390769333169797567531, −10.5956865214858219664313658337, −9.26156361733504028806682540772, −7.92995700561787230340331596707, −7.071690689622795646235600533066, −6.36930965172830323920642974387, −4.807859866523655353819001196778, −2.81385201588684978180930212261, −0.989058669872707697076693945223, 0.63250730457688003909520175905, 3.15316727744210636583095633095, 3.84420294147884546820039602003, 5.70993363919087444167633861289, 6.90784378781887942576359817601, 8.40142087562849324496069080615, 9.18310473546257118186035696232, 10.4671343106893796086380349463, 11.204208312786966839015750171712, 12.087079999001563028333704647854, 13.22863017224942904541969538429, 15.47564381192731585642489456517, 15.82624805446092220821895637109, 16.61621974681333974661302993564, 17.80877540379772945345992088581, 18.905673112779225688272215296619, 19.685235867632758676756610337475, 20.75005742333872879647874342145, 21.68566971196716899387328543827, 22.66218355195793686766659735282, 23.63727414968169147579130422531, 24.945233395288180433364985073176, 26.346973638899759422969549771457, 26.66146501903256196574848349192, 27.730989618594707810883126996074

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