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Properties

Label 1-209-209.27-r1-0-0
Degree $1$
Conductor $209$
Sign $-0.777 - 0.629i$
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  + (−0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.777 - 0.629i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ -0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01988224458 + 0.05616600202i\)
\(L(\frac12)\) \(\approx\) \(0.01988224458 + 0.05616600202i\)
\(L(1)\) \(\approx\) \(0.4732518257 + 0.1112245338i\)
\(L(1)\) \(\approx\) \(0.4732518257 + 0.1112245338i\)

Euler product

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

−25.6350297231888959223932605549, −24.68065426129907029433540860785, −24.56050565513351322496081968094, −23.406511239493551788411500587777, −22.39025877669647198615831701278, −20.930131403484865968511811103918, −20.02419061837316710122472932933, −19.02159501989382782592496767977, −18.05179382457039707329200752833, −17.570964275865890114753149291381, −16.35034185942367843113144953238, −15.81952467405151566471405934308, −14.47180512532727006414795583417, −13.023433471685187298322906188377, −12.08542439336180315554431197615, −11.29185610902688703254445801690, −9.999502408161344128221576957661, −8.68943307555755845550036922607, −8.09336892159827935660531575417, −6.88772212119806935618275730629, −5.624284427435088625954632197301, −5.06908527993303975969143882932, −2.43278217915163248804615658218, −1.19485973845258333201135720755, −0.03258381255015955005991113578, 1.692153307038022886827864062884, 3.43548770690978372369596274250, 4.242825269878162768626550875335, 6.19400447886289730306052910578, 7.038410866793740025490441181549, 8.23626260946186096520351602032, 9.72865998396125217190797042689, 10.296373092154448987223302636519, 11.282271087288475296337031054537, 11.76117413561208955368851913653, 13.47147030456454037102925709845, 14.82418003905853245277902694833, 15.732566170058559112299025082971, 16.89923189884747348674511048429, 17.382184687871589779091425965349, 18.4561331867271059808791009365, 19.38767992245418107274949760947, 20.46679537032829411528227201410, 21.411771999433346508662925868048, 22.08836217062129708884192647770, 23.243106195930439310451573864928, 24.106297374432952510606240215211, 25.84190594832658946586069880332, 26.37035630349324619559433057598, 26.99058462642059809547342999353

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