Properties

Label 2-1944-216.43-c0-0-1
Degree $2$
Conductor $1944$
Sign $-0.448 - 0.893i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.500 + 0.866i)8-s + (−1.43 + 0.524i)11-s + (−0.939 + 0.342i)16-s + (0.939 + 1.62i)17-s + (−0.173 + 0.300i)19-s + (−1.43 − 0.524i)22-s + (0.766 + 0.642i)25-s + (−0.939 − 0.342i)32-s + (−0.326 + 1.85i)34-s + (−0.326 + 0.118i)38-s + (0.266 − 0.223i)41-s + (1.76 − 0.642i)43-s + (−0.766 − 1.32i)44-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.500 + 0.866i)8-s + (−1.43 + 0.524i)11-s + (−0.939 + 0.342i)16-s + (0.939 + 1.62i)17-s + (−0.173 + 0.300i)19-s + (−1.43 − 0.524i)22-s + (0.766 + 0.642i)25-s + (−0.939 − 0.342i)32-s + (−0.326 + 1.85i)34-s + (−0.326 + 0.118i)38-s + (0.266 − 0.223i)41-s + (1.76 − 0.642i)43-s + (−0.766 − 1.32i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.448 - 0.893i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542710830\)
\(L(\frac12)\) \(\approx\) \(1.542710830\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.568014686279940411658052639629, −8.529161060140867547337167169628, −7.892619999945132250679425370082, −7.35065714939848186153781785102, −6.33022766157915950753174748220, −5.59574750818720292826508148695, −4.93241644630640544412886096179, −3.94533710061696132756330164745, −3.08093391364812346600425015509, −1.98329943019660303178165381658, 0.880761588254465777176600955243, 2.59575344926785343145080441703, 2.93715750199233864845552312308, 4.20744685159412530731377526403, 5.12610921671157832061077036995, 5.57893203631644462957605568821, 6.63080297655535878647514745409, 7.49272207104592012218743431377, 8.344565407491048673003847220759, 9.432871347566459421053345805051

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