↑

Properties

Label 1-223-223.43-r0-0-0
Degree $1$
Conductor $223$
Sign $-0.993 - 0.112i$
Analytic cond. $1.03560$
Root an. cond. $1.03560$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.450 − 0.892i)2-s + (−0.980 − 0.196i)3-s + (−0.594 + 0.803i)4-s + (−0.346 + 0.938i)5-s + (0.265 + 0.964i)6-s + (−0.721 − 0.691i)7-s + (0.985 + 0.169i)8-s + (0.922 + 0.385i)9-s + (0.993 − 0.112i)10-s + (0.702 − 0.712i)11-s + (0.741 − 0.671i)12-s + (0.660 + 0.750i)13-s + (−0.292 + 0.956i)14-s + (0.524 − 0.851i)15-s + (−0.292 − 0.956i)16-s + (−0.967 − 0.251i)17-s + ⋯
L(s)  = 1  + (−0.450 − 0.892i)2-s + (−0.980 − 0.196i)3-s + (−0.594 + 0.803i)4-s + (−0.346 + 0.938i)5-s + (0.265 + 0.964i)6-s + (−0.721 − 0.691i)7-s + (0.985 + 0.169i)8-s + (0.922 + 0.385i)9-s + (0.993 − 0.112i)10-s + (0.702 − 0.712i)11-s + (0.741 − 0.671i)12-s + (0.660 + 0.750i)13-s + (−0.292 + 0.956i)14-s + (0.524 − 0.851i)15-s + (−0.292 − 0.956i)16-s + (−0.967 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(223\)
Sign: $-0.993 - 0.112i$
Analytic conductor: \(1.03560\)
Root analytic conductor: \(1.03560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{223} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 223,\ (0:\ ),\ -0.993 - 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01336364080 - 0.2360024265i\)
\(L(\frac12)\) \(\approx\) \(0.01336364080 - 0.2360024265i\)
\(L(1)\) \(\approx\) \(0.4016966905 - 0.1991226608i\)
\(L(1)\) \(\approx\) \(0.4016966905 - 0.1991226608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad223 \( 1 \)
good2 \( 1 + (-0.450 - 0.892i)T \)
3 \( 1 + (-0.980 - 0.196i)T \)
5 \( 1 + (-0.346 + 0.938i)T \)
7 \( 1 + (-0.721 - 0.691i)T \)
11 \( 1 + (0.702 - 0.712i)T \)
13 \( 1 + (0.660 + 0.750i)T \)
17 \( 1 + (-0.967 - 0.251i)T \)
19 \( 1 + (-0.886 + 0.462i)T \)
23 \( 1 + (-0.639 - 0.769i)T \)
29 \( 1 + (0.974 + 0.224i)T \)
31 \( 1 + (-0.858 - 0.512i)T \)
37 \( 1 + (0.265 - 0.964i)T \)
41 \( 1 + (-0.911 + 0.411i)T \)
43 \( 1 + (-0.182 - 0.983i)T \)
47 \( 1 + (-0.886 - 0.462i)T \)
53 \( 1 + (-0.990 - 0.141i)T \)
59 \( 1 + (0.210 - 0.977i)T \)
61 \( 1 + (0.319 - 0.947i)T \)
67 \( 1 + (0.993 + 0.112i)T \)
71 \( 1 + (-0.951 + 0.306i)T \)
73 \( 1 + (0.424 - 0.905i)T \)
79 \( 1 + (0.0988 - 0.995i)T \)
83 \( 1 + (-0.182 + 0.983i)T \)
89 \( 1 + (-0.346 - 0.938i)T \)
97 \( 1 + (-0.639 + 0.769i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.22778686247308308921458594428, −25.76952289567088148672312390608, −25.103871779585876893004375278476, −24.10268598821378386553039881283, −23.34237304194615202103007420794, −22.52850924032481171497140985866, −21.65006754942978388420172983865, −20.097948018173687770882347010856, −19.30784345740967106561713477945, −18.00463755099035188777694695426, −17.42066818439603051083578345922, −16.441902587898617218039751669007, −15.66250147066390726637852729508, −15.14566553761655545860983740698, −13.27812252661222385748205072889, −12.569604251446349137109163314380, −11.40480646916635964400596892187, −10.10262379281941748860779221597, −9.202977049359599916426515596177, −8.2942396097173314143045353506, −6.80950714514951280893562416766, −6.04777625410028347572482704623, −5.00005659236585466311949586702, −4.044401119190893818454869457965, −1.42263047491931461081401669632, 0.24387339434474171687892732393, 1.935816823787691592217570205504, 3.57035894150251071678515528725, 4.30890631483980401929623002588, 6.34465876672840173363789289478, 6.9106014257161833967621500753, 8.3399667296440499338097265633, 9.71144509629157552255329772658, 10.747642597782986216858588016894, 11.19483058329909931540441779222, 12.20016428366724025545535998164, 13.29322169365729237831152470381, 14.20420074413555363378134754467, 16.01996474334627998149571231672, 16.66494625063914390139076273609, 17.694009876001704408503201538693, 18.66550407679033927532241182694, 19.194848015965957060046802910680, 20.20331021582748205191785205368, 21.65976459406851844122130415084, 22.16918477381783365552362680612, 23.04104529292762191530469200568, 23.74770074349948735170087330678, 25.33178527169468556344652037787, 26.50276462894106635571162992524

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