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Properties

Label 2-1944-72.5-c0-0-7
Degree $2$
Conductor $1944$
Sign $0.766 + 0.642i$
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 + (0.173 − 0.300i)5-s + (−0.766 − 1.32i)7-s − 0.999·8-s + 0.347·10-s + (−0.939 − 1.62i)11-s + (0.766 − 1.32i)14-s + (−0.5 − 0.866i)16-s + (0.173 + 0.300i)20-s + (0.939 − 1.62i)22-s + (0.439 + 0.761i)25-s + 1.53·28-s + (−0.5 − 0.866i)29-s + (0.939 − 1.62i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.173 − 0.300i)5-s + (−0.766 − 1.32i)7-s − 0.999·8-s + 0.347·10-s + (−0.939 − 1.62i)11-s + (0.766 − 1.32i)14-s + (−0.5 − 0.866i)16-s + (0.173 + 0.300i)20-s + (0.939 − 1.62i)22-s + (0.439 + 0.761i)25-s + 1.53·28-s + (−0.5 − 0.866i)29-s + (0.939 − 1.62i)31-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.766 + 0.642i$
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: \((2,\ 1944,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9859804789\)
\(L(\frac12)\) \(\approx\) \(0.9859804789\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
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 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.234180938577843898138190469519, −8.155640185640534904681992268415, −7.79487303185727685559035980822, −6.83208090546927293320497928413, −6.09419487687288052404464501479, −5.44567297101561848570194415011, −4.40151477930745776442608210580, −3.60348534104017486889612388554, −2.81824913247301014345314706263, −0.58705155651037440442586064987, 1.86896385989886679699449880677, 2.63169482751191600476980552697, 3.33977755036969985046519585983, 4.75348512521887233286301166854, 5.15905688986330527049751137305, 6.21934902185453086544026653889, 6.82508465351054552223901643558, 8.050858381893853370421717988767, 9.030736914036466804911666034187, 9.573862398896913312440187635524

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