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Properties

Label 1-223-223.206-r1-0-0
Degree $1$
Conductor $223$
Sign $-0.0955 + 0.995i$
Analytic cond. $23.9646$
Root an. cond. $23.9646$
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  + (0.0424 − 0.999i)2-s + (0.594 + 0.803i)3-s + (−0.996 − 0.0848i)4-s + (0.127 + 0.991i)5-s + (0.828 − 0.559i)6-s + (0.210 + 0.977i)7-s + (−0.127 + 0.991i)8-s + (−0.292 + 0.956i)9-s + (0.996 − 0.0848i)10-s + (0.828 − 0.559i)11-s + (−0.524 − 0.851i)12-s + (0.594 − 0.803i)13-s + (0.985 − 0.169i)14-s + (−0.721 + 0.691i)15-s + (0.985 + 0.169i)16-s + (−0.828 + 0.559i)17-s + ⋯
L(s)  = 1  + (0.0424 − 0.999i)2-s + (0.594 + 0.803i)3-s + (−0.996 − 0.0848i)4-s + (0.127 + 0.991i)5-s + (0.828 − 0.559i)6-s + (0.210 + 0.977i)7-s + (−0.127 + 0.991i)8-s + (−0.292 + 0.956i)9-s + (0.996 − 0.0848i)10-s + (0.828 − 0.559i)11-s + (−0.524 − 0.851i)12-s + (0.594 − 0.803i)13-s + (0.985 − 0.169i)14-s + (−0.721 + 0.691i)15-s + (0.985 + 0.169i)16-s + (−0.828 + 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0955 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0955 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(223\)
Sign: $-0.0955 + 0.995i$
Analytic conductor: \(23.9646\)
Root analytic conductor: \(23.9646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{223} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 223,\ (1:\ ),\ -0.0955 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188868704 + 1.308437650i\)
\(L(\frac12)\) \(\approx\) \(1.188868704 + 1.308437650i\)
\(L(1)\) \(\approx\) \(1.175792168 + 0.2229673002i\)
\(L(1)\) \(\approx\) \(1.175792168 + 0.2229673002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad223 \( 1 \)
good2 \( 1 + (0.0424 - 0.999i)T \)
3 \( 1 + (0.594 + 0.803i)T \)
5 \( 1 + (0.127 + 0.991i)T \)
7 \( 1 + (0.210 + 0.977i)T \)
11 \( 1 + (0.828 - 0.559i)T \)
13 \( 1 + (0.594 - 0.803i)T \)
17 \( 1 + (-0.828 + 0.559i)T \)
19 \( 1 + (-0.911 - 0.411i)T \)
23 \( 1 + (0.127 + 0.991i)T \)
29 \( 1 + (0.985 + 0.169i)T \)
31 \( 1 + (0.372 + 0.927i)T \)
37 \( 1 + (-0.828 - 0.559i)T \)
41 \( 1 + (-0.450 + 0.892i)T \)
43 \( 1 + (-0.967 - 0.251i)T \)
47 \( 1 + (-0.911 + 0.411i)T \)
53 \( 1 + (0.778 - 0.628i)T \)
59 \( 1 + (-0.524 + 0.851i)T \)
61 \( 1 + (0.594 - 0.803i)T \)
67 \( 1 + (0.996 + 0.0848i)T \)
71 \( 1 + (-0.524 + 0.851i)T \)
73 \( 1 + (0.660 - 0.750i)T \)
79 \( 1 + (0.450 - 0.892i)T \)
83 \( 1 + (-0.967 + 0.251i)T \)
89 \( 1 + (-0.127 + 0.991i)T \)
97 \( 1 + (0.127 - 0.991i)T \)
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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

−25.78953691714833277174937779163, −24.92066446709469922127047320587, −24.29530767701273827650246240718, −23.524218743422598878065254901530, −22.742067365267517003783510844877, −21.15897364726118244625972225424, −20.29641866072765181504816470108, −19.37224406376270816423820402928, −18.25915206513999179386160531498, −17.22514625571606146863126633194, −16.74631014114151657075572901865, −15.46176376726986842251205311728, −14.297128807744298566951251508623, −13.67002670840004052546139347871, −12.86807998762151039330587726275, −11.811442359473428285773808805952, −9.909355767546442871380299328576, −8.81390297267711231126629663264, −8.26316331264013986006080058937, −6.950262378458374555916025211864, −6.388816352570297836004133545506, −4.64574320119290232110785250163, −3.93057568573055409210091504078, −1.729699199926963195448159804397, −0.53830877217903929875782253812, 1.848679272873546831912249872631, 2.95937760520493884192908736025, 3.69666124791002715300330906126, 5.06497516588384148178016024082, 6.29778365760684396754445032765, 8.32019598696749769823512847914, 8.881515286368968204958789494369, 10.05730611413687585988731901724, 10.90464347167378331638425490131, 11.62012611642587782582294627111, 13.09187399314737684725412071489, 14.0515575142154761860756070445, 14.91398385156114436715286321276, 15.63841980686339882162915461314, 17.35040813790521944458428877492, 18.183488492967528535996486136306, 19.36506877137797706287520744325, 19.699773600966206144035363258783, 21.19638826135680883249633049578, 21.65857095057887175133914320122, 22.29324291452434540923575807591, 23.29061085390598756201798909698, 24.9115125424601517880870506418, 25.76535253384071529153117755292, 26.70854873865280030322922426167

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