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Properties

Label 2-1944-1944.1195-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.0645 + 0.997i$
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  + (0.466 − 0.884i)2-s + (−0.981 − 0.192i)3-s + (−0.565 − 0.824i)4-s + (−0.627 + 0.778i)6-s + (−0.993 + 0.116i)8-s + (0.925 + 0.378i)9-s + (1.70 + 1.03i)11-s + (0.396 + 0.918i)12-s + (−0.360 + 0.932i)16-s + (0.424 − 1.41i)17-s + (0.766 − 0.642i)18-s + (0.0377 + 0.00894i)19-s + (1.70 − 1.03i)22-s + (0.996 + 0.0774i)24-s + (0.952 + 0.305i)25-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)2-s + (−0.981 − 0.192i)3-s + (−0.565 − 0.824i)4-s + (−0.627 + 0.778i)6-s + (−0.993 + 0.116i)8-s + (0.925 + 0.378i)9-s + (1.70 + 1.03i)11-s + (0.396 + 0.918i)12-s + (−0.360 + 0.932i)16-s + (0.424 − 1.41i)17-s + (0.766 − 0.642i)18-s + (0.0377 + 0.00894i)19-s + (1.70 − 1.03i)22-s + (0.996 + 0.0774i)24-s + (0.952 + 0.305i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0645 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0645 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.0645 + 0.997i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.0645 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101164550\)
\(L(\frac12)\) \(\approx\) \(1.101164550\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.466 + 0.884i)T \)
3 \( 1 + (0.981 + 0.192i)T \)
good5 \( 1 + (-0.952 - 0.305i)T^{2} \)
7 \( 1 + (-0.323 + 0.946i)T^{2} \)
11 \( 1 + (-1.70 - 1.03i)T + (0.466 + 0.884i)T^{2} \)
13 \( 1 + (-0.0968 - 0.995i)T^{2} \)
17 \( 1 + (-0.424 + 1.41i)T + (-0.835 - 0.549i)T^{2} \)
19 \( 1 + (-0.0377 - 0.00894i)T + (0.893 + 0.448i)T^{2} \)
23 \( 1 + (0.981 + 0.192i)T^{2} \)
29 \( 1 + (-0.249 - 0.968i)T^{2} \)
31 \( 1 + (-0.856 - 0.516i)T^{2} \)
37 \( 1 + (-0.396 + 0.918i)T^{2} \)
41 \( 1 + (-0.345 + 0.547i)T + (-0.431 - 0.902i)T^{2} \)
43 \( 1 + (1.04 + 0.205i)T + (0.925 + 0.378i)T^{2} \)
47 \( 1 + (-0.856 + 0.516i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.0139 + 0.720i)T + (-0.999 + 0.0387i)T^{2} \)
61 \( 1 + (0.360 + 0.932i)T^{2} \)
67 \( 1 + (1.28 - 0.996i)T + (0.249 - 0.968i)T^{2} \)
71 \( 1 + (0.686 - 0.727i)T^{2} \)
73 \( 1 + (0.161 + 0.217i)T + (-0.286 + 0.957i)T^{2} \)
79 \( 1 + (-0.996 + 0.0774i)T^{2} \)
83 \( 1 + (-0.952 - 1.51i)T + (-0.431 + 0.902i)T^{2} \)
89 \( 1 + (-0.770 + 1.78i)T + (-0.686 - 0.727i)T^{2} \)
97 \( 1 + (1.90 - 0.297i)T + (0.952 - 0.305i)T^{2} \)
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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

−9.543849048962633436197725297570, −8.697843866397019051275666018397, −7.20215217690791345748221321159, −6.78881809175580617239515830324, −5.81583576693369013972778330720, −4.95188606679585785291123289660, −4.37447600990088172172395830433, −3.37587809787910982074984141084, −2.00762108693367147201331069011, −1.02313463253135801641108865917, 1.22744776147458492947223805672, 3.31417185074094656704239468809, 4.02207220628253934666897577695, 4.79637803661852687596209591666, 5.88405783302475435363731208448, 6.23583686643171488184885954018, 6.88512252025433814779176448688, 7.913718168051539849733354406228, 8.764883235118530552880260635663, 9.365570440753191618307100096433

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