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Properties

Label 2-1944-1944.1363-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.180 + 0.983i$
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  + (0.323 − 0.946i)2-s + (−0.431 + 0.902i)3-s + (−0.790 − 0.612i)4-s + (0.713 + 0.700i)6-s + (−0.835 + 0.549i)8-s + (−0.627 − 0.778i)9-s + (−0.220 − 0.157i)11-s + (0.893 − 0.448i)12-s + (0.249 + 0.968i)16-s + (1.73 − 0.203i)17-s + (−0.939 + 0.342i)18-s + (−0.721 − 1.67i)19-s + (−0.220 + 0.157i)22-s + (−0.135 − 0.990i)24-s + (0.856 − 0.516i)25-s + ⋯
L(s)  = 1  + (0.323 − 0.946i)2-s + (−0.431 + 0.902i)3-s + (−0.790 − 0.612i)4-s + (0.713 + 0.700i)6-s + (−0.835 + 0.549i)8-s + (−0.627 − 0.778i)9-s + (−0.220 − 0.157i)11-s + (0.893 − 0.448i)12-s + (0.249 + 0.968i)16-s + (1.73 − 0.203i)17-s + (−0.939 + 0.342i)18-s + (−0.721 − 1.67i)19-s + (−0.220 + 0.157i)22-s + (−0.135 − 0.990i)24-s + (0.856 − 0.516i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9930538091\)
\(L(\frac12)\) \(\approx\) \(0.9930538091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.323 + 0.946i)T \)
3 \( 1 + (0.431 - 0.902i)T \)
good5 \( 1 + (-0.856 + 0.516i)T^{2} \)
7 \( 1 + (0.565 - 0.824i)T^{2} \)
11 \( 1 + (0.220 + 0.157i)T + (0.323 + 0.946i)T^{2} \)
13 \( 1 + (-0.533 + 0.845i)T^{2} \)
17 \( 1 + (-1.73 + 0.203i)T + (0.973 - 0.230i)T^{2} \)
19 \( 1 + (0.721 + 1.67i)T + (-0.686 + 0.727i)T^{2} \)
23 \( 1 + (0.431 - 0.902i)T^{2} \)
29 \( 1 + (0.740 - 0.672i)T^{2} \)
31 \( 1 + (-0.813 - 0.581i)T^{2} \)
37 \( 1 + (-0.893 - 0.448i)T^{2} \)
41 \( 1 + (-1.11 - 0.217i)T + (0.925 + 0.378i)T^{2} \)
43 \( 1 + (-0.846 + 1.77i)T + (-0.627 - 0.778i)T^{2} \)
47 \( 1 + (-0.813 + 0.581i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.454 + 0.206i)T + (0.657 + 0.753i)T^{2} \)
61 \( 1 + (-0.249 + 0.968i)T^{2} \)
67 \( 1 + (0.335 - 0.869i)T + (-0.740 - 0.672i)T^{2} \)
71 \( 1 + (-0.597 + 0.802i)T^{2} \)
73 \( 1 + (0.114 + 1.97i)T + (-0.993 + 0.116i)T^{2} \)
79 \( 1 + (0.135 - 0.990i)T^{2} \)
83 \( 1 + (-1.34 + 0.264i)T + (0.925 - 0.378i)T^{2} \)
89 \( 1 + (-0.707 - 0.355i)T + (0.597 + 0.802i)T^{2} \)
97 \( 1 + (1.83 + 0.510i)T + (0.856 + 0.516i)T^{2} \)
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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

−9.271470738118655648216863645489, −8.918517988987084179393882030263, −7.81552782616385898187437847765, −6.54817098871198843768695615299, −5.67857140932295433726862306278, −4.98596724877735928184372167294, −4.28430589644664723347026268812, −3.30776965475133251229082511000, −2.55791291129902894329628032011, −0.797787168087831561651414219755, 1.31518843512577745305179240258, 2.85865246495491923985330263523, 3.90427128574645401355885594356, 5.03351870887833571911209656187, 5.76223732408536044270698265910, 6.28403012573649188961667738282, 7.22311405680298193124216145155, 7.901168588504743422590609198603, 8.276480891909929224753225816228, 9.403189498427775882554084479930

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