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Properties

Label 2-209-19.7-c3-0-16
Degree $2$
Conductor $209$
Sign $-0.900 + 0.435i$
Analytic cond. $12.3313$
Root an. cond. $3.51160$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.42 + 4.20i)2-s + (4.62 + 8.01i)3-s + (−7.79 + 13.5i)4-s + (−2.98 − 5.17i)5-s + (−22.4 + 38.9i)6-s + 11.7·7-s − 36.9·8-s + (−29.3 + 50.7i)9-s + (14.5 − 25.1i)10-s + 11·11-s − 144.·12-s + (14.8 − 25.7i)13-s + (28.4 + 49.3i)14-s + (27.6 − 47.8i)15-s + (−27.2 − 47.2i)16-s + (−41.4 − 71.7i)17-s + ⋯
L(s)  = 1  + (0.858 + 1.48i)2-s + (0.890 + 1.54i)3-s + (−0.974 + 1.68i)4-s + (−0.267 − 0.462i)5-s + (−1.52 + 2.64i)6-s + 0.633·7-s − 1.63·8-s + (−1.08 + 1.88i)9-s + (0.458 − 0.794i)10-s + 0.301·11-s − 3.47·12-s + (0.317 − 0.549i)13-s + (0.543 + 0.942i)14-s + (0.475 − 0.823i)15-s + (−0.426 − 0.738i)16-s + (−0.590 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(12.3313\)
Root analytic conductor: \(3.51160\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :3/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.769503 - 3.35722i\)
\(L(\frac12)\) \(\approx\) \(0.769503 - 3.35722i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - 11T \)
19 \( 1 + (-11.5 - 82.0i)T \)
good2 \( 1 + (-2.42 - 4.20i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.62 - 8.01i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (2.98 + 5.17i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 11.7T + 343T^{2} \)
13 \( 1 + (-14.8 + 25.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (41.4 + 71.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-12.4 + 21.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (16.0 - 27.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 - 303.T + 5.06e4T^{2} \)
41 \( 1 + (-149. - 258. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (59.2 + 102. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-136. + 235. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (229. - 397. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (304. + 528. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (226. - 392. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (71.9 - 124. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-111. - 192. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (282. + 490. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (517. + 895. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.25e3T + 5.71e5T^{2} \)
89 \( 1 + (-785. + 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-646. - 1.11e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.94907011142447164200314756288, −11.58359261970800237194485359078, −10.31770328659059032799218720868, −9.114676042767843911024665905605, −8.356675979118252200307598734823, −7.65535243005516565099853529987, −6.00340813621409675052132044955, −4.76963820283854608237209140790, −4.39386539641580490861000142666, −3.13033985872223191084645953328, 1.13815537876549205658544650157, 2.13802154340074538107103952524, 3.15224921418871498014792034729, 4.38749211349325333648769055639, 6.15432073239063128453173656301, 7.26978975358666425371575323268, 8.432793799180457710642832737292, 9.439984591357212211121517242378, 11.04511258708265896529529214235, 11.46187483102759511027647191630

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