↑

Properties

Label 2-3344-209.170-c0-0-2
Degree $2$
Conductor $3344$
Sign $0.162 + 0.986i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
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  + (0.564 − 1.73i)5-s + (−0.169 + 0.122i)7-s + (0.309 + 0.951i)9-s + (0.104 − 0.994i)11-s + (0.413 − 1.27i)17-s + (0.809 + 0.587i)19-s − 0.618·23-s + (−1.89 − 1.37i)25-s + (0.118 + 0.363i)35-s + 1.95·43-s + 1.82·45-s + (−1.58 − 1.14i)47-s + (−0.295 + 0.909i)49-s + (−1.66 − 0.743i)55-s + (−0.604 + 1.86i)61-s + ⋯
L(s)  = 1  + (0.564 − 1.73i)5-s + (−0.169 + 0.122i)7-s + (0.309 + 0.951i)9-s + (0.104 − 0.994i)11-s + (0.413 − 1.27i)17-s + (0.809 + 0.587i)19-s − 0.618·23-s + (−1.89 − 1.37i)25-s + (0.118 + 0.363i)35-s + 1.95·43-s + 1.82·45-s + (−1.58 − 1.14i)47-s + (−0.295 + 0.909i)49-s + (−1.66 − 0.743i)55-s + (−0.604 + 1.86i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395215300\)
\(L(\frac12)\) \(\approx\) \(1.395215300\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.169 - 0.122i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.95T + T^{2} \)
47 \( 1 + (1.58 + 1.14i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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

−8.663656001963431627043644590524, −7.995029654777199287884073632718, −7.38530345753323899414166489402, −6.07037531503952893281814195218, −5.53196438392092901200349922290, −4.94075139674917858618692185147, −4.16904850440559008395210139354, −2.98670809487861076203281368029, −1.84288683103493075561003017204, −0.883477306127772288050237485292, 1.58541816070951151400099808557, 2.56763002867147891264684756258, 3.44521552914975097941369749498, 4.06668672052398050563478436396, 5.32853840713244766250660345962, 6.39965503994306374985860496255, 6.48845586734461087629093120262, 7.38662090683472571000252332214, 7.973941262071117702480855202490, 9.359367684828533779996554394202

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