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Properties

Label 2-209-11.3-c1-0-8
Degree $2$
Conductor $209$
Sign $0.783 - 0.621i$
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  + (−1.63 + 1.19i)2-s + (−0.0193 − 0.0594i)3-s + (0.649 − 1.99i)4-s + (2.71 + 1.97i)5-s + (0.102 + 0.0743i)6-s + (1.46 − 4.52i)7-s + (0.0630 + 0.193i)8-s + (2.42 − 1.76i)9-s − 6.79·10-s + (−3.30 + 0.217i)11-s − 0.131·12-s + (1.05 − 0.763i)13-s + (2.97 + 9.15i)14-s + (0.0647 − 0.199i)15-s + (3.06 + 2.22i)16-s + (4.38 + 3.18i)17-s + ⋯
L(s)  = 1  + (−1.15 + 0.841i)2-s + (−0.0111 − 0.0343i)3-s + (0.324 − 0.998i)4-s + (1.21 + 0.881i)5-s + (0.0417 + 0.0303i)6-s + (0.555 − 1.70i)7-s + (0.0222 + 0.0685i)8-s + (0.807 − 0.587i)9-s − 2.14·10-s + (−0.997 + 0.0657i)11-s − 0.0378·12-s + (0.291 − 0.211i)13-s + (0.795 + 2.44i)14-s + (0.0167 − 0.0514i)15-s + (0.766 + 0.556i)16-s + (1.06 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.783 - 0.621i$
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: \((2,\ 209,\ (\ :1/2),\ 0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817744 + 0.284815i\)
\(L(\frac12)\) \(\approx\) \(0.817744 + 0.284815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.30 - 0.217i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (1.63 - 1.19i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.0193 + 0.0594i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.71 - 1.97i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.46 + 4.52i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.05 + 0.763i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.38 - 3.18i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 0.858T + 23T^{2} \)
29 \( 1 + (1.16 - 3.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.33 - 3.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.820 - 2.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.77 + 5.47i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 + (-1.08 - 3.34i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.4 + 8.33i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.57 - 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.324 - 0.236i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.04T + 67T^{2} \)
71 \( 1 + (-3.88 - 2.82i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.271 - 0.835i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.71 - 2.69i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.549 - 0.399i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + (4.25 - 3.09i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.73761118204443731362373884779, −10.74703694143285354750612819965, −10.30502442810555689885367722408, −9.767806568809626306697403732957, −8.320811744946289416794641504711, −7.27974575275394346677826847385, −6.82649723635722764637038003230, −5.56828915469593333264528765072, −3.67203730792591072597166400904, −1.36632250759405931190079861296, 1.65829323474117100825581952468, 2.48547147455348404444521161356, 5.15432554411803325778656337034, 5.65949177137846660659285649448, 7.82199334668292808268022970555, 8.601664801116505014057373483265, 9.502271916405108181826347084273, 10.00479480147896675545678645726, 11.16836867146415760901798458939, 12.15115553696511670544763414612

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