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Properties

Label 2-209-19.7-c1-0-4
Degree $2$
Conductor $209$
Sign $-0.839 - 0.542i$
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  + (1.31 + 2.28i)2-s + (0.631 + 1.09i)3-s + (−2.47 + 4.28i)4-s + (−0.759 − 1.31i)5-s + (−1.66 + 2.88i)6-s + 1.94·7-s − 7.77·8-s + (0.702 − 1.21i)9-s + (2.00 − 3.46i)10-s − 11-s − 6.25·12-s + (1.57 − 2.72i)13-s + (2.56 + 4.44i)14-s + (0.959 − 1.66i)15-s + (−5.30 − 9.18i)16-s + (0.682 + 1.18i)17-s + ⋯
L(s)  = 1  + (0.932 + 1.61i)2-s + (0.364 + 0.631i)3-s + (−1.23 + 2.14i)4-s + (−0.339 − 0.588i)5-s + (−0.679 + 1.17i)6-s + 0.735·7-s − 2.75·8-s + (0.234 − 0.405i)9-s + (0.633 − 1.09i)10-s − 0.301·11-s − 1.80·12-s + (0.436 − 0.756i)13-s + (0.685 + 1.18i)14-s + (0.247 − 0.429i)15-s + (−1.32 − 2.29i)16-s + (0.165 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.839 - 0.542i$
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: \((2,\ 209,\ (\ :1/2),\ -0.839 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545549 + 1.84959i\)
\(L(\frac12)\) \(\approx\) \(0.545549 + 1.84959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.37 - 2.76i)T \)
good2 \( 1 + (-1.31 - 2.28i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.631 - 1.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.759 + 1.31i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.94T + 7T^{2} \)
13 \( 1 + (-1.57 + 2.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.682 - 1.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.194 - 0.337i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.909 - 1.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + (5.02 + 8.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.249 - 0.431i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.73 - 9.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.490 - 0.849i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.98 + 5.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.22 + 3.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.90 + 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.53 + 6.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.27 - 7.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.58 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.70T + 83T^{2} \)
89 \( 1 + (0.833 - 1.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.72 + 15.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.81907311483737245924164326380, −12.46500432071863253801361239391, −10.89427921606373653120011526202, −9.383380137576990565149768541548, −8.322130661772869500969674245713, −7.88494780418990910548984297064, −6.47546339454052288909248350123, −5.34986550424181611025106557601, −4.42642598237463640529431996861, −3.54504113836738133443230260641, 1.66086715866769182826346391818, 2.74266725772947154777400046843, 4.11189680633162856379740932501, 5.15075898883313816316522547562, 6.72433135493316502458404438548, 8.054386482025759236674284035419, 9.323869190216158921255019613953, 10.56757022337613064817594335859, 11.20774388977915604109300089092, 11.86261274947850173889958684614

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