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Properties

Label 2-209-19.11-c1-0-8
Degree $2$
Conductor $209$
Sign $0.466 - 0.884i$
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  + (−0.615 + 1.06i)2-s + (1.01 − 1.75i)3-s + (0.241 + 0.417i)4-s + (−1.31 + 2.27i)5-s + (1.24 + 2.16i)6-s + 3.75·7-s − 3.05·8-s + (−0.555 − 0.962i)9-s + (−1.62 − 2.80i)10-s − 11-s + 0.978·12-s + (0.824 + 1.42i)13-s + (−2.31 + 4.00i)14-s + (2.66 + 4.62i)15-s + (1.40 − 2.42i)16-s + (0.706 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.435 + 0.754i)2-s + (0.585 − 1.01i)3-s + (0.120 + 0.208i)4-s + (−0.588 + 1.01i)5-s + (0.509 + 0.883i)6-s + 1.42·7-s − 1.08·8-s + (−0.185 − 0.320i)9-s + (−0.512 − 0.887i)10-s − 0.301·11-s + 0.282·12-s + (0.228 + 0.396i)13-s + (−0.618 + 1.07i)14-s + (0.688 + 1.19i)15-s + (0.350 − 0.606i)16-s + (0.171 − 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.466 - 0.884i$
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: \((2,\ 209,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04314 + 0.629207i\)
\(L(\frac12)\) \(\approx\) \(1.04314 + 0.629207i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.29 - 0.756i)T \)
good2 \( 1 + (0.615 - 1.06i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.01 + 1.75i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.31 - 2.27i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.75T + 7T^{2} \)
13 \( 1 + (-0.824 - 1.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.706 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.21 - 2.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.60 + 6.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
37 \( 1 + 9.68T + 37T^{2} \)
41 \( 1 + (-6.11 + 10.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.96 + 5.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0433 + 0.0750i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.599 - 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.36 + 2.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.87 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.07 + 3.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.31 - 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.11 - 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.58 - 2.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.23T + 83T^{2} \)
89 \( 1 + (-7.60 - 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.66 + 2.87i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.45955051931341631924776393458, −11.57975278081843056288355504824, −10.85376931083171953204749780588, −9.107099427090597252845020317792, −8.112917393058000103696042170415, −7.37834369787984073207245460700, −7.15360851495513252783197712459, −5.53224914056613876681536081486, −3.53386260860338388534078763862, −2.10016784650197640496502675543, 1.38658539029827566677919895529, 3.25099244481248862893182803798, 4.57475651232126319194076607416, 5.45624772389531176816688122049, 7.62081768612596837170486411727, 8.702991232161491008548080678627, 9.149980527245543426968420478402, 10.37196574718643269460931658264, 11.04971354888269063655894759781, 11.94072977010332620142985934068

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