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Properties

Label 2-209-209.189-c1-0-7
Degree $2$
Conductor $209$
Sign $0.745 + 0.666i$
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  + (0.410 − 1.26i)2-s + (0.190 + 0.138i)4-s + (0.5 + 1.53i)5-s + (−0.690 + 0.951i)7-s + (2.40 − 1.74i)8-s + (0.927 − 2.85i)9-s + 2.14·10-s + (2.80 + 1.76i)11-s + (0.820 − 2.52i)13-s + (0.917 + 1.26i)14-s + (−1.07 − 3.30i)16-s + (−5.42 + 1.76i)17-s + (−3.22 − 2.34i)18-s + (−3.05 + 3.11i)19-s + (−0.118 + 0.363i)20-s + ⋯
L(s)  = 1  + (0.290 − 0.893i)2-s + (0.0954 + 0.0693i)4-s + (0.223 + 0.688i)5-s + (−0.261 + 0.359i)7-s + (0.849 − 0.617i)8-s + (0.309 − 0.951i)9-s + 0.679·10-s + (0.846 + 0.531i)11-s + (0.227 − 0.700i)13-s + (0.245 + 0.337i)14-s + (−0.268 − 0.825i)16-s + (−1.31 + 0.427i)17-s + (−0.759 − 0.552i)18-s + (−0.699 + 0.714i)19-s + (−0.0263 + 0.0812i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48819 - 0.568036i\)
\(L(\frac12)\) \(\approx\) \(1.48819 - 0.568036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.80 - 1.76i)T \)
19 \( 1 + (3.05 - 3.11i)T \)
good2 \( 1 + (-0.410 + 1.26i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.5 - 1.53i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.690 - 0.951i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.820 + 2.52i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (5.42 - 1.76i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 + 2.61T + 23T^{2} \)
29 \( 1 + (5.62 + 4.08i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.80 - 1.56i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.80 - 6.61i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.77 - 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.15iT - 43T^{2} \)
47 \( 1 + (-3.11 + 2.26i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (12.5 + 4.08i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.83 + 2.52i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.954 + 0.310i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 + (-10.7 + 3.49i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.47 + 6.15i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.83 + 5.64i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (10.5 - 3.44i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.05iT - 89T^{2} \)
97 \( 1 + (2.96 + 0.964i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.25131269551701996158824527083, −11.41595600729992932964305215276, −10.40237607784633126019441049628, −9.715250476690307273444057004067, −8.402390061678681355043716630283, −6.85774938035578827955082785845, −6.29350100296194851069772313028, −4.27798782470744409969394846359, −3.29270869548593780114806225609, −1.91673874122289620673786149154, 1.88771909185065441128849951203, 4.24509104374568934199834681345, 5.15883588476752373997049637574, 6.43666504138237789954883724892, 7.13352789791993831291837704470, 8.437826290239317167656679869527, 9.289092773103113553233290091005, 10.73723010434097364513625357092, 11.34855695847461560841808146685, 12.83585870164019680952791766879

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