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Properties

Label 2-209-19.7-c1-0-7
Degree $2$
Conductor $209$
Sign $-0.0305 - 0.999i$
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  + (0.527 + 0.913i)2-s + (1.63 + 2.82i)3-s + (0.444 − 0.769i)4-s + (−0.650 − 1.12i)5-s + (−1.71 + 2.97i)6-s − 1.25·7-s + 3.04·8-s + (−3.81 + 6.61i)9-s + (0.685 − 1.18i)10-s − 11-s + 2.89·12-s + (1.93 − 3.35i)13-s + (−0.663 − 1.14i)14-s + (2.12 − 3.67i)15-s + (0.717 + 1.24i)16-s + (−1.18 − 2.04i)17-s + ⋯
L(s)  = 1  + (0.372 + 0.645i)2-s + (0.941 + 1.63i)3-s + (0.222 − 0.384i)4-s + (−0.290 − 0.503i)5-s + (−0.701 + 1.21i)6-s − 0.475·7-s + 1.07·8-s + (−1.27 + 2.20i)9-s + (0.216 − 0.375i)10-s − 0.301·11-s + 0.836·12-s + (0.537 − 0.931i)13-s + (−0.177 − 0.307i)14-s + (0.547 − 0.948i)15-s + (0.179 + 0.310i)16-s + (−0.286 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0305 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0305 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.0305 - 0.999i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ -0.0305 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29425 + 1.33442i\)
\(L(\frac12)\) \(\approx\) \(1.29425 + 1.33442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
19 \( 1 + (4.18 + 1.20i)T \)
good2 \( 1 + (-0.527 - 0.913i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.63 - 2.82i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.650 + 1.12i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.25T + 7T^{2} \)
13 \( 1 + (-1.93 + 3.35i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.18 + 2.04i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.15 - 3.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.65 + 2.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.36T + 31T^{2} \)
37 \( 1 + 5.09T + 37T^{2} \)
41 \( 1 + (-4.57 - 7.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.197 + 0.341i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.96 - 6.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.53 + 2.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.47 - 4.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 4.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.76 - 9.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.11 - 5.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.05 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.08 + 7.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.25T + 83T^{2} \)
89 \( 1 + (6.84 - 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 + 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.07519392937981279087401315034, −11.35047759663293703180627678469, −10.37985557150660320480388064427, −9.804874868368438777078901312302, −8.606488477295600511919004510449, −7.87301275095357028098077727549, −6.18820983550336588480752799549, −4.99969215915800703157811563785, −4.23616298515513956871654827704, −2.82773492377216269158167461450, 1.84691336472297986664942694817, 2.88373896513593931220845070858, 3.93488303148906376057095073136, 6.45477079012796477034243492229, 6.97089107996010904803869313884, 8.088293463824753957905203318693, 8.790181103860632152129643522258, 10.46258873352729358874403291546, 11.54622693721076788014213992556, 12.36728079932571318652645739925

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