↑

Properties

Label 2-209-19.11-c1-0-10
Degree $2$
Conductor $209$
Sign $0.986 - 0.165i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 − 0.776i)2-s + (−0.820 + 1.42i)3-s + (0.597 + 1.03i)4-s + (1.31 − 2.27i)5-s + (0.735 + 1.27i)6-s − 1.27·7-s + 2.86·8-s + (0.154 + 0.266i)9-s + (−1.17 − 2.04i)10-s + 11-s − 1.96·12-s + (2.92 + 5.06i)13-s + (−0.571 + 0.989i)14-s + (2.15 + 3.73i)15-s + (0.0888 − 0.153i)16-s + (0.838 − 1.45i)17-s + ⋯
L(s)  = 1  + (0.317 − 0.549i)2-s + (−0.473 + 0.820i)3-s + (0.298 + 0.517i)4-s + (0.588 − 1.01i)5-s + (0.300 + 0.520i)6-s − 0.481·7-s + 1.01·8-s + (0.0513 + 0.0889i)9-s + (−0.373 − 0.646i)10-s + 0.301·11-s − 0.566·12-s + (0.810 + 1.40i)13-s + (−0.152 + 0.264i)14-s + (0.557 + 0.965i)15-s + (0.0222 − 0.0384i)16-s + (0.203 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.986 - 0.165i$
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: \((2,\ 209,\ (\ :1/2),\ 0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45483 + 0.121062i\)
\(L(\frac12)\) \(\approx\) \(1.45483 + 0.121062i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 + 3.87i)T \)
good2 \( 1 + (-0.448 + 0.776i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.820 - 1.42i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.31 + 2.27i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
13 \( 1 + (-2.92 - 5.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.838 + 1.45i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.35 + 2.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 + 0.294T + 37T^{2} \)
41 \( 1 + (-3.15 + 5.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.38 + 4.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.76 + 4.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.90 - 11.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.53 - 9.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.11 + 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.54 + 2.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.83 - 4.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.14 + 3.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.84 + 4.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 + (-1.29 - 2.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.82 - 11.8i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33719770011592890503959178804, −11.42013469457160603099897886541, −10.68229718825850531707194360788, −9.487624504566328485182835551222, −8.828333874562977658394107069841, −7.25205278657731707426921853582, −5.92551614852203824674668262111, −4.63497576539202016290376920325, −3.89588600251694915729523717457, −1.98144422269969581766730086032, 1.60073509334439255275348329949, 3.46343297857179618127761766082, 5.63226784740629568443061272926, 6.15396082206317707944947315670, 6.90382758437604559795336974179, 7.87996070463475843678883949550, 9.657066751029797172932940076241, 10.54914443832244655175848672969, 11.23215479190855444721629039172, 12.69557921149338002325291663901

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