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Properties

Label 2-209-19.7-c1-0-8
Degree $2$
Conductor $209$
Sign $0.836 - 0.548i$
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  + (0.797 + 1.38i)2-s + (−0.583 − 1.01i)3-s + (−0.273 + 0.472i)4-s + (0.206 + 0.357i)5-s + (0.931 − 1.61i)6-s + 2.20·7-s + 2.31·8-s + (0.819 − 1.41i)9-s + (−0.329 + 0.571i)10-s − 11-s + 0.637·12-s + (−2.41 + 4.18i)13-s + (1.75 + 3.04i)14-s + (0.241 − 0.417i)15-s + (2.39 + 4.15i)16-s + (0.827 + 1.43i)17-s + ⋯
L(s)  = 1  + (0.564 + 0.977i)2-s + (−0.336 − 0.583i)3-s + (−0.136 + 0.236i)4-s + (0.0924 + 0.160i)5-s + (0.380 − 0.658i)6-s + 0.833·7-s + 0.820·8-s + (0.273 − 0.472i)9-s + (−0.104 + 0.180i)10-s − 0.301·11-s + 0.183·12-s + (−0.670 + 1.16i)13-s + (0.470 + 0.814i)14-s + (0.0622 − 0.107i)15-s + (0.599 + 1.03i)16-s + (0.200 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.836 - 0.548i$
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: \((2,\ 209,\ (\ :1/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56571 + 0.467216i\)
\(L(\frac12)\) \(\approx\) \(1.56571 + 0.467216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 4.20i)T \)
good2 \( 1 + (-0.797 - 1.38i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.583 + 1.01i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.206 - 0.357i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.20T + 7T^{2} \)
13 \( 1 + (2.41 - 4.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.827 - 1.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.50 - 4.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.74 + 4.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.790 + 1.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.20 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.812 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.46 - 4.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.71 + 13.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.30 - 10.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.75 - 6.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.84 + 6.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.46T + 83T^{2} \)
89 \( 1 + (1.80 - 3.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.45 - 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.58923330115669318115418747715, −11.66957175402455332745008778542, −10.69976659721377336668304462445, −9.426505382019407873364562774459, −8.013846793219819220386505709597, −7.10754349892206652072413933812, −6.41224196104063565706180188981, −5.25847666007441476902338718116, −4.22711203434635698606285355393, −1.82544433245107939078958293956, 1.92608041727284611156244102108, 3.45729148815063599369126255051, 4.79772126626815792353780152878, 5.33753302869977487434561358442, 7.38475283442285310925920654224, 8.251314905072405453603287270638, 9.892465193966334030451102846736, 10.61513147438329776032677053764, 11.18203304820951562791066067371, 12.39593502519627087875379823694

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