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Properties

Label 1-209-209.7-r1-0-0
Degree $1$
Conductor $209$
Sign $0.971 + 0.238i$
Analytic cond. $22.4601$
Root an. cond. $22.4601$
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  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.978 − 0.207i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 + 0.866i)10-s + 12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.978 − 0.207i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 + 0.866i)10-s + 12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ 0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.744327785 + 0.6961791732i\)
\(L(\frac12)\) \(\approx\) \(5.744327785 + 0.6961791732i\)
\(L(1)\) \(\approx\) \(2.992945245 + 0.2530889336i\)
\(L(1)\) \(\approx\) \(2.992945245 + 0.2530889336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−26.175283125069891150646294234977, −25.2253027000614466620015006643, −24.533254198941484302748504426273, −23.98798938791903150747803245552, −22.2678320429848569190519730758, −21.67489434695884842678707883516, −20.87584529968329321180385572892, −20.190112600656627895206413964292, −19.31012901800181443106174359836, −17.84040817732968924753496265376, −16.64724966238270130433336749173, −15.40497373935629488297358151307, −14.887153174166412751965018639492, −13.80681782254919964193852220822, −13.05592794984795604582240139124, −12.07691316026760286046653611160, −10.73345970602602065125947421690, −9.712553519362439841881741241915, −8.58776907847707290552130298847, −7.50371822604471402436186000831, −5.76073784266751090733995717570, −4.981069970797497912604248080694, −3.91727377214122067298477836019, −2.44726150214445206652734870466, −1.66862748235917034433508038800, 1.767615381826971145707508795461, 2.57693636214830107398493430008, 3.87504795894488437456916391795, 5.01871293377420447542374274974, 6.64724334589563490153667638008, 7.15325725920499607171866252756, 8.33539979739728573553549800940, 9.80245388092215251493866094029, 10.98918051534156046401187152333, 12.082764722951614685714151808479, 13.34592350578779160516894917053, 14.07925302548861088048989219562, 14.51339253590970693803557779230, 15.53740669182234161477800630430, 16.9558013659934201516576733293, 17.93561696246931059572479580520, 19.05856651432757758812667548744, 20.22767999236970122994282133850, 20.89628928538460519085802128652, 21.797396450200489565081406855427, 22.73942285339350542142910287835, 23.99375428938162259777541276727, 24.48533724672743556618446039835, 25.441023897403137746830800966012, 26.36396708739079781445925941288

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