↑

Properties

Label 2-209-11.3-c1-0-12
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

Related objects

Downloads

Learn more

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: $\chi_{209} (58, \cdot )$
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 \( 1 + (8.23 - 5.98i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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