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Properties

Label 2-1944-216.43-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8332557991\)
\(L(\frac12)\) \(\approx\) \(0.8332557991\)
\(L(1)\) 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

−9.190917039159174240372289422600, −8.800394288660724122637879426163, −7.80630588575525963064662616843, −6.94414754804195591248980978512, −6.41036851483030959281705222126, −5.04403419144909412456493830945, −4.06211706192589180833096946910, −3.23333530720737685700273102437, −2.18308043809103179629519509987, −0.927744116007820154604292628073, 1.29331507943810206805392541623, 2.32582936610298164701426781261, 3.96599278655358906830206531083, 4.64999547900774370734978514284, 5.86348078283716674875021458191, 6.53818535490269463452461473836, 7.02742206228320145518454656549, 8.114538972635365403784752332781, 8.716488227627083326058179119758, 9.338321496788474660956889936466

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