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Properties

Label 2-209-11.9-c1-0-7
Degree $2$
Conductor $209$
Sign $0.780 + 0.625i$
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  + (−0.690 + 2.12i)2-s + (−2.61 + 1.90i)3-s + (−2.42 − 1.76i)4-s + (−0.5 − 1.53i)5-s + (−2.23 − 6.88i)6-s + (0.309 + 0.224i)7-s + (1.80 − 1.31i)8-s + (2.30 − 7.10i)9-s + 3.61·10-s + (−2.19 + 2.48i)11-s + 9.70·12-s + (−0.763 + 2.35i)13-s + (−0.690 + 0.502i)14-s + (4.23 + 3.07i)15-s + (−0.309 − 0.951i)16-s + (−0.954 − 2.93i)17-s + ⋯
L(s)  = 1  + (−0.488 + 1.50i)2-s + (−1.51 + 1.09i)3-s + (−1.21 − 0.881i)4-s + (−0.223 − 0.688i)5-s + (−0.912 − 2.80i)6-s + (0.116 + 0.0848i)7-s + (0.639 − 0.464i)8-s + (0.769 − 2.36i)9-s + 1.14·10-s + (−0.660 + 0.750i)11-s + 2.80·12-s + (−0.211 + 0.652i)13-s + (−0.184 + 0.134i)14-s + (1.09 + 0.794i)15-s + (−0.0772 − 0.237i)16-s + (−0.231 − 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.780 + 0.625i)\)

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 \( 1 + (2.19 - 2.48i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.690 - 2.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.61 - 1.90i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.5 + 1.53i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.309 - 0.224i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.763 - 2.35i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.954 + 2.93i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + 0.618T + 23T^{2} \)
29 \( 1 + (6.85 + 4.97i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.47 + 7.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.61 - 1.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.47 - 6.88i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.09T + 43T^{2} \)
47 \( 1 + (6.97 - 5.06i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.472 + 1.45i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.0 + 7.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.66 - 8.19i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.47T + 67T^{2} \)
71 \( 1 + (-3.61 - 11.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.85 + 2.07i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.909 + 2.80i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.64 - 5.06i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.23T + 89T^{2} \)
97 \( 1 + (3.76 - 11.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.00588481941670517396862723375, −11.32661684733918876811927596365, −9.893962218427811955782134487575, −9.470670100269648658155903151682, −8.188923870162329040989399260081, −6.96406251284040542215911937405, −6.00734318140263758546301502102, −4.99918362663808183979678570279, −4.46307322620676663816582169035, 0, 1.64573595739454665205560995993, 3.19437350227552603502290507400, 5.11088394603245845419789530394, 6.33190479101790687539674362111, 7.42683046167353182119910172175, 8.550844396920578934606165787139, 10.33311030885369749133081267530, 10.77353049691927471923718887362, 11.32750098911099306928542418926

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