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Properties

Label 2-209-19.11-c1-0-12
Degree $2$
Conductor $209$
Sign $0.845 + 0.534i$
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  + (−0.0694 + 0.120i)2-s + (0.748 − 1.29i)3-s + (0.990 + 1.71i)4-s + (1.75 − 3.04i)5-s + (0.103 + 0.179i)6-s − 1.46·7-s − 0.552·8-s + (0.379 + 0.657i)9-s + (0.244 + 0.422i)10-s − 11-s + 2.96·12-s + (−1.19 − 2.07i)13-s + (0.101 − 0.176i)14-s + (−2.63 − 4.55i)15-s + (−1.94 + 3.36i)16-s + (1.62 − 2.80i)17-s + ⋯
L(s)  = 1  + (−0.0490 + 0.0850i)2-s + (0.432 − 0.748i)3-s + (0.495 + 0.857i)4-s + (0.786 − 1.36i)5-s + (0.0424 + 0.0734i)6-s − 0.555·7-s − 0.195·8-s + (0.126 + 0.219i)9-s + (0.0772 + 0.133i)10-s − 0.301·11-s + 0.855·12-s + (−0.331 − 0.574i)13-s + (0.0272 − 0.0472i)14-s + (−0.679 − 1.17i)15-s + (−0.485 + 0.841i)16-s + (0.392 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47345 - 0.426477i\)
\(L(\frac12)\) \(\approx\) \(1.47345 - 0.426477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 4.22i)T \)
good2 \( 1 + (0.0694 - 0.120i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.748 + 1.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.75 + 3.04i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.46T + 7T^{2} \)
13 \( 1 + (1.19 + 2.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.62 + 2.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.69 - 6.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.20 + 5.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.866T + 31T^{2} \)
37 \( 1 + 2.78T + 37T^{2} \)
41 \( 1 + (4.92 - 8.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.65 - 6.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.67 + 2.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.89 - 3.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.20 - 7.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.718 - 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.24 + 5.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.70 + 2.95i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.40 - 2.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.85 + 15.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + (6.43 + 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.86 + 13.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.56195959610403738879505766991, −11.71909070172864031343387838052, −10.12107715390254869744165541683, −9.182152490816246564853076951052, −8.101510075364194136527822605373, −7.50777599526868171275936855449, −6.16596556918202143820499902329, −4.93735901672846743512611870063, −3.12225138242825851872560699139, −1.70939794489446019063378583610, 2.29304530911603406488447543962, 3.38533802087740995271757086415, 5.13670549679476856935241465549, 6.55547250320376770373367772903, 6.87737467881718996342260341429, 8.938611812982199169288625158907, 9.772808063422296310345557484253, 10.46231570414199819393008184762, 10.97267591150302607523758608945, 12.43758122197334734254146990657

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