Properties

Label 2-209-209.197-c0-0-1
Degree $2$
Conductor $209$
Sign $0.305 + 0.952i$
Analytic cond. $0.104304$
Root an. cond. $0.322962$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.499i)6-s + i·8-s + (−0.866 − 0.499i)10-s + i·11-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s i·19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.866 − 0.499i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.499i)6-s + i·8-s + (−0.866 − 0.499i)10-s + i·11-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s i·19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.866 − 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(0.104304\)
Root analytic conductor: \(0.322962\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :0),\ 0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8350514765\)
\(L(\frac12)\) \(\approx\) \(0.8350514765\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - iT \)
19 \( 1 + iT \)
good2 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−12.59878214374348844857471046594, −11.70170822259938823687102300683, −11.15214454433226423544601240658, −9.384598762562672584106446621976, −8.356505921760740358469166939759, −7.26017384000536900167833545953, −6.05319430970867544832352090710, −4.73452519105335723026706593379, −3.89465344068019022949133309138, −1.88468753944685974228414804407, 3.39534326667322651269387170485, 4.26725185241816332100587755687, 5.51610305641054235868624484660, 6.28222701480822907240696897913, 7.49654642368921679881128986129, 8.896500537691307310573059846531, 10.29690649426814426782369016975, 10.77478702258525130717597088530, 11.71555922436764316537261612012, 13.08864861527252854924438333706

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