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Properties

Label 2-209-19.7-c3-0-11
Degree $2$
Conductor $209$
Sign $0.806 - 0.591i$
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  + (−2.56 − 4.43i)2-s + (4.17 + 7.22i)3-s + (−9.11 + 15.7i)4-s + (1.66 + 2.88i)5-s + (21.3 − 37.0i)6-s + 27.0·7-s + 52.3·8-s + (−21.3 + 36.9i)9-s + (8.54 − 14.7i)10-s − 11·11-s − 152.·12-s + (−7.81 + 13.5i)13-s + (−69.3 − 120. i)14-s + (−13.9 + 24.1i)15-s + (−61.2 − 106. i)16-s + (−18.1 − 31.4i)17-s + ⋯
L(s)  = 1  + (−0.905 − 1.56i)2-s + (0.803 + 1.39i)3-s + (−1.13 + 1.97i)4-s + (0.149 + 0.258i)5-s + (1.45 − 2.51i)6-s + 1.46·7-s + 2.31·8-s + (−0.790 + 1.36i)9-s + (0.270 − 0.467i)10-s − 0.301·11-s − 3.66·12-s + (−0.166 + 0.288i)13-s + (−1.32 − 2.29i)14-s + (−0.239 + 0.414i)15-s + (−0.957 − 1.65i)16-s + (−0.259 − 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.806 - 0.591i$
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: \((2,\ 209,\ (\ :3/2),\ 0.806 - 0.591i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34695 + 0.441151i\)
\(L(\frac12)\) \(\approx\) \(1.34695 + 0.441151i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-26.2 - 78.5i)T \)
good2 \( 1 + (2.56 + 4.43i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.17 - 7.22i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-1.66 - 2.88i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 27.0T + 343T^{2} \)
13 \( 1 + (7.81 - 13.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (18.1 + 31.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (55.7 - 96.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (111. - 193. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 47.4T + 2.97e4T^{2} \)
37 \( 1 - 61.7T + 5.06e4T^{2} \)
41 \( 1 + (61.4 + 106. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-205. - 355. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (242. - 420. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-73.9 + 128. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (379. + 657. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-450. + 780. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (505. - 875. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-133. - 230. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-177. - 307. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-518. - 898. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 237.T + 5.71e5T^{2} \)
89 \( 1 + (-131. + 227. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (738. + 1.27e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.41740786034405585203787381572, −10.96543868214041255868162345277, −10.02984762377160965625118819135, −9.414545738182532556003035587941, −8.440333440314945052586769119288, −7.78939843681009455662321095258, −5.00779127502806744954853178709, −4.03447612424420460098734072468, −2.91419276322772587718602806577, −1.71301685536772804885035001822, 0.78398053495826439628510674030, 2.07027590121547852511088200807, 4.83465623828024028101429129029, 5.99700324331676850720019676614, 7.15964817330020497403346198220, 7.77942419862303706768178346047, 8.450305556792498209660381611929, 9.135350852354844452162668759458, 10.58659040197930919000098404145, 11.95328214400305635071640593167

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