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Properties

Label 2-209-11.3-c1-0-0
Degree $2$
Conductor $209$
Sign $-0.660 + 0.750i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 0.936i)2-s + (0.935 + 2.87i)3-s + (0.166 − 0.511i)4-s + (−2.91 − 2.11i)5-s + (−3.90 − 2.83i)6-s + (0.0705 − 0.217i)7-s + (−0.719 − 2.21i)8-s + (−4.98 + 3.61i)9-s + 5.74·10-s + (−3.21 + 0.802i)11-s + 1.62·12-s + (−1.03 + 0.753i)13-s + (0.112 + 0.345i)14-s + (3.37 − 10.3i)15-s + (3.87 + 2.81i)16-s + (5.03 + 3.65i)17-s + ⋯
L(s)  = 1  + (−0.911 + 0.662i)2-s + (0.539 + 1.66i)3-s + (0.0831 − 0.255i)4-s + (−1.30 − 0.947i)5-s + (−1.59 − 1.15i)6-s + (0.0266 − 0.0820i)7-s + (−0.254 − 0.783i)8-s + (−1.66 + 1.20i)9-s + 1.81·10-s + (−0.970 + 0.242i)11-s + 0.470·12-s + (−0.287 + 0.209i)13-s + (0.0300 + 0.0924i)14-s + (0.870 − 2.67i)15-s + (0.968 + 0.703i)16-s + (1.22 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.660 + 0.750i$
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: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ -0.660 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144097 - 0.318825i\)
\(L(\frac12)\) \(\approx\) \(0.144097 - 0.318825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.21 - 0.802i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (1.28 - 0.936i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.935 - 2.87i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.91 + 2.11i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0705 + 0.217i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.03 - 0.753i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.03 - 3.65i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 7.86T + 23T^{2} \)
29 \( 1 + (-0.00257 + 0.00791i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.07 - 4.41i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.24 + 3.82i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.87 - 5.76i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.42T + 43T^{2} \)
47 \( 1 + (0.516 + 1.58i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.124 + 0.0902i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.70 + 5.23i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.70 - 5.59i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + (-3.08 - 2.24i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.11 - 6.52i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.87 + 6.44i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.136 - 0.0989i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.97T + 89T^{2} \)
97 \( 1 + (-0.0190 + 0.0138i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.82701912707860302798441598162, −11.95194207340306658749266016817, −10.55529756326075435262975299750, −9.860146960597007733833334055602, −8.916108242829931740715091314693, −8.118639824527110591514646610299, −7.66471480565363960355127297392, −5.50366175334161531231603335943, −4.27705939552346878204024169958, −3.53237525374661053024848961624, 0.36430052517561302172726081913, 2.29030345221075216978373069044, 3.24601747575296114073070847422, 5.74188382817937322479465567258, 7.27685381709797315370975265312, 7.76594054998996381906054626785, 8.433539664110856089907361930907, 9.846840075920903888485957897785, 10.94358467143805940793968075795, 11.82432771271426749023855879819

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