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Properties

Label 2-209-11.4-c1-0-9
Degree $2$
Conductor $209$
Sign $-0.859 - 0.511i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 1.31i)2-s + (−0.381 + 1.17i)3-s + (0.927 + 2.85i)4-s + (−0.5 + 0.363i)5-s + (2.23 − 1.62i)6-s + (−0.809 − 2.48i)7-s + (0.690 − 2.12i)8-s + (1.19 + 0.865i)9-s + 1.38·10-s + (−3.30 + 0.224i)11-s − 3.70·12-s + (−5.23 − 3.80i)13-s + (−1.80 + 5.56i)14-s + (−0.236 − 0.726i)15-s + (0.809 − 0.587i)16-s + (−6.54 + 4.75i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.929i)2-s + (−0.220 + 0.678i)3-s + (0.463 + 1.42i)4-s + (−0.223 + 0.162i)5-s + (0.912 − 0.663i)6-s + (−0.305 − 0.941i)7-s + (0.244 − 0.751i)8-s + (0.396 + 0.288i)9-s + 0.437·10-s + (−0.997 + 0.0676i)11-s − 1.07·12-s + (−1.45 − 1.05i)13-s + (−0.483 + 1.48i)14-s + (−0.0609 − 0.187i)15-s + (0.202 − 0.146i)16-s + (−1.58 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.30 - 0.224i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (1.80 + 1.31i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.381 - 1.17i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.5 - 0.363i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.809 + 2.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.23 + 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.54 - 4.75i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 1.61T + 23T^{2} \)
29 \( 1 + (0.145 + 0.449i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.47 + 4.70i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.618 + 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.527 - 1.62i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.09T + 43T^{2} \)
47 \( 1 + (-1.97 + 6.06i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.47 + 6.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.09 - 3.35i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.16 - 3.75i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.472T + 67T^{2} \)
71 \( 1 + (-1.38 + 1.00i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.85 - 11.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.0 - 8.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.35 + 6.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.23T + 89T^{2} \)
97 \( 1 + (8.23 + 5.98i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.15990266202560784863169216684, −10.67084627301868355807922335392, −10.08926692306780838852888685430, −9.242656680116865933603382336535, −7.87167680529517951508725982607, −7.25775108490073092197607341953, −5.21000985753832792481343059233, −3.83920409701706473591518737828, −2.32448512207610311663395453394, 0, 2.26621755709725202451702479990, 4.85507114474475453256001817807, 6.24402811132466963744407242095, 7.07508515094084491870994891969, 7.76954187880331326595654398905, 9.108117522287294934077140897734, 9.443855428670256303445776106738, 10.80673845427158191134578460255, 12.10456154347071232241440698061

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