↑

Properties

Label 2-209-19.7-c1-0-14
Degree $2$
Conductor $209$
Sign $0.242 + 0.970i$
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  + (1.00 + 1.74i)2-s + (−1.29 − 2.23i)3-s + (−1.02 + 1.77i)4-s + (−1.77 − 3.07i)5-s + (2.59 − 4.50i)6-s − 4.09·7-s − 0.111·8-s + (−1.82 + 3.16i)9-s + (3.57 − 6.19i)10-s − 11-s + 5.30·12-s + (2.55 − 4.42i)13-s + (−4.12 − 7.14i)14-s + (−4.58 + 7.93i)15-s + (1.94 + 3.36i)16-s + (0.559 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.711 + 1.23i)2-s + (−0.744 − 1.29i)3-s + (−0.513 + 0.889i)4-s + (−0.794 − 1.37i)5-s + (1.06 − 1.83i)6-s − 1.54·7-s − 0.0393·8-s + (−0.609 + 1.05i)9-s + (1.13 − 1.95i)10-s − 0.301·11-s + 1.53·12-s + (0.708 − 1.22i)13-s + (−1.10 − 1.91i)14-s + (−1.18 + 2.04i)15-s + (0.485 + 0.841i)16-s + (0.135 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.242 + 0.970i$
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: \((2,\ 209,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711288 - 0.555598i\)
\(L(\frac12)\) \(\approx\) \(0.711288 - 0.555598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.34 - 0.288i)T \)
good2 \( 1 + (-1.00 - 1.74i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.29 + 2.23i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.09T + 7T^{2} \)
13 \( 1 + (-2.55 + 4.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.629 + 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.20 + 5.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.563T + 31T^{2} \)
37 \( 1 + 1.00T + 37T^{2} \)
41 \( 1 + (2.90 + 5.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.48 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.10 - 5.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.68 - 4.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.01 - 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.778 + 1.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.65 + 6.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.30 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.84T + 83T^{2} \)
89 \( 1 + (-0.742 + 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.33 - 14.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.50297726101297305571511755911, −11.87483760312201092792962192598, −10.25640477507787364075172487618, −8.615146900706273945723489935305, −7.77407574578923115205169856717, −6.92075366123956759879878028819, −5.91172210847437745618766481465, −5.28220824918710337904748437548, −3.69711655077345180306563399532, −0.68879155200787775693656748897, 3.11481419293209081964884584851, 3.55068197365047054171469118555, 4.67110312099220426664490383384, 6.15168597703119968137403203861, 7.21216651807741353374990309190, 9.363342330556938292907271739234, 10.12431394420828933306305405309, 10.82465159777444628898558580128, 11.46783059890474748169995753284, 12.15164119525714233589130289284

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