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Properties

Label 2-1944-1944.1843-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.896 - 0.443i$
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  + (0.533 + 0.845i)2-s + (0.657 − 0.753i)3-s + (−0.431 + 0.902i)4-s + (0.987 + 0.154i)6-s + (−0.993 + 0.116i)8-s + (−0.135 − 0.990i)9-s + (0.990 − 0.546i)11-s + (0.396 + 0.918i)12-s + (−0.627 − 0.778i)16-s + (0.121 − 0.405i)17-s + (0.766 − 0.642i)18-s + (1.66 + 0.394i)19-s + (0.990 + 0.546i)22-s + (−0.565 + 0.824i)24-s + (−0.740 + 0.672i)25-s + ⋯
L(s)  = 1  + (0.533 + 0.845i)2-s + (0.657 − 0.753i)3-s + (−0.431 + 0.902i)4-s + (0.987 + 0.154i)6-s + (−0.993 + 0.116i)8-s + (−0.135 − 0.990i)9-s + (0.990 − 0.546i)11-s + (0.396 + 0.918i)12-s + (−0.627 − 0.778i)16-s + (0.121 − 0.405i)17-s + (0.766 − 0.642i)18-s + (1.66 + 0.394i)19-s + (0.990 + 0.546i)22-s + (−0.565 + 0.824i)24-s + (−0.740 + 0.672i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.854676247\)
\(L(\frac12)\) \(\approx\) \(1.854676247\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.533 - 0.845i)T \)
3 \( 1 + (-0.657 + 0.753i)T \)
good5 \( 1 + (0.740 - 0.672i)T^{2} \)
7 \( 1 + (0.981 - 0.192i)T^{2} \)
11 \( 1 + (-0.990 + 0.546i)T + (0.533 - 0.845i)T^{2} \)
13 \( 1 + (0.910 + 0.413i)T^{2} \)
17 \( 1 + (-0.121 + 0.405i)T + (-0.835 - 0.549i)T^{2} \)
19 \( 1 + (-1.66 - 0.394i)T + (0.893 + 0.448i)T^{2} \)
23 \( 1 + (-0.657 + 0.753i)T^{2} \)
29 \( 1 + (-0.713 + 0.700i)T^{2} \)
31 \( 1 + (0.875 - 0.483i)T^{2} \)
37 \( 1 + (-0.396 + 0.918i)T^{2} \)
41 \( 1 + (-1.96 - 0.0760i)T + (0.996 + 0.0774i)T^{2} \)
43 \( 1 + (1.31 - 1.50i)T + (-0.135 - 0.990i)T^{2} \)
47 \( 1 + (0.875 + 0.483i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (1.07 - 0.648i)T + (0.466 - 0.884i)T^{2} \)
61 \( 1 + (0.627 - 0.778i)T^{2} \)
67 \( 1 + (-0.179 - 0.0732i)T + (0.713 + 0.700i)T^{2} \)
71 \( 1 + (0.686 - 0.727i)T^{2} \)
73 \( 1 + (0.943 + 1.26i)T + (-0.286 + 0.957i)T^{2} \)
79 \( 1 + (0.565 + 0.824i)T^{2} \)
83 \( 1 + (1.78 - 0.0693i)T + (0.996 - 0.0774i)T^{2} \)
89 \( 1 + (-0.770 + 1.78i)T + (-0.686 - 0.727i)T^{2} \)
97 \( 1 + (0.179 - 0.465i)T + (-0.740 - 0.672i)T^{2} \)
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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

−9.334382396684429996667719307783, −8.427846546422464277298872071360, −7.67383776436433331494178354300, −7.21806778369200937393215996697, −6.23900325671054004329012125883, −5.74370253857038870389185695883, −4.54885460130175295888604810810, −3.50850294563580648612562847264, −2.95682047455484600220995230550, −1.35374978950215379366888689132, 1.52328744460882763869341953767, 2.60150612270269715154439965521, 3.54929119281696520967528043674, 4.17812307096672178451361141656, 5.01705172533489380131811860976, 5.82907629439453761045725005525, 6.92432334186565667938392912613, 7.946847337193907279102068780246, 8.882447553072090052698335497987, 9.562495905620177868461172186860

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