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Properties

Label 2-1944-72.5-c0-0-8
Degree $2$
Conductor $1944$
Sign $-0.939 + 0.342i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)5-s + (−0.173 − 0.300i)7-s + 0.999·8-s − 1.87·10-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)14-s + (−0.5 − 0.866i)16-s + (0.939 + 1.62i)20-s + (−0.766 + 1.32i)22-s + (−1.26 − 2.19i)25-s + 0.347·28-s + (0.5 + 0.866i)29-s + (−0.766 + 1.32i)31-s + (−0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)5-s + (−0.173 − 0.300i)7-s + 0.999·8-s − 1.87·10-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)14-s + (−0.5 − 0.866i)16-s + (0.939 + 1.62i)20-s + (−0.766 + 1.32i)22-s + (−1.26 − 2.19i)25-s + 0.347·28-s + (0.5 + 0.866i)29-s + (−0.766 + 1.32i)31-s + (−0.499 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8649456606\)
\(L(\frac12)\) \(\approx\) \(0.8649456606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.347T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.957809527184730855124680009232, −8.587362996417991848517590861382, −7.915369005983864583974981903288, −6.69466352475734249840606708872, −5.46329345248428459339540128308, −5.07070323869175982043831632470, −3.95035057621037754001821971209, −2.92692556572061106053707352515, −1.74808370792006120066403392121, −0.74801962818237825558376684117, 1.97801602805078371994154140742, 2.65092140129290858932888959127, 4.10287485093965554084532949888, 5.26329843606893522449638940224, 5.97357168718337061467038189460, 6.62128279898116464800277992636, 7.36611498712631163927456200090, 7.83749000412173996934744211299, 9.126984019791900148130113248423, 9.728151563141538731232372256192

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