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Properties

Label 2-209-19.7-c1-0-5
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: $\chi_{209} (45, \cdot )$
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

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

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