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Properties

Label 2-1944-1944.1291-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.971 - 0.236i$
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  + (0.996 − 0.0774i)2-s + (−0.790 + 0.612i)3-s + (0.987 − 0.154i)4-s + (−0.740 + 0.672i)6-s + (0.973 − 0.230i)8-s + (0.249 − 0.968i)9-s + (1.25 + 0.0486i)11-s + (−0.686 + 0.727i)12-s + (0.952 − 0.305i)16-s + (−1.35 − 0.894i)17-s + (0.173 − 0.984i)18-s + (0.952 + 0.478i)19-s + (1.25 − 0.0486i)22-s + (−0.627 + 0.778i)24-s + (−0.910 + 0.413i)25-s + ⋯
L(s)  = 1  + (0.996 − 0.0774i)2-s + (−0.790 + 0.612i)3-s + (0.987 − 0.154i)4-s + (−0.740 + 0.672i)6-s + (0.973 − 0.230i)8-s + (0.249 − 0.968i)9-s + (1.25 + 0.0486i)11-s + (−0.686 + 0.727i)12-s + (0.952 − 0.305i)16-s + (−1.35 − 0.894i)17-s + (0.173 − 0.984i)18-s + (0.952 + 0.478i)19-s + (1.25 − 0.0486i)22-s + (−0.627 + 0.778i)24-s + (−0.910 + 0.413i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.823924624\)
\(L(\frac12)\) \(\approx\) \(1.823924624\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 + 0.0774i)T \)
3 \( 1 + (0.790 - 0.612i)T \)
good5 \( 1 + (0.910 - 0.413i)T^{2} \)
7 \( 1 + (0.135 - 0.990i)T^{2} \)
11 \( 1 + (-1.25 - 0.0486i)T + (0.996 + 0.0774i)T^{2} \)
13 \( 1 + (-0.323 + 0.946i)T^{2} \)
17 \( 1 + (1.35 + 0.894i)T + (0.396 + 0.918i)T^{2} \)
19 \( 1 + (-0.952 - 0.478i)T + (0.597 + 0.802i)T^{2} \)
23 \( 1 + (0.790 - 0.612i)T^{2} \)
29 \( 1 + (-0.856 - 0.516i)T^{2} \)
31 \( 1 + (0.999 + 0.0387i)T^{2} \)
37 \( 1 + (0.686 + 0.727i)T^{2} \)
41 \( 1 + (-0.153 + 0.223i)T + (-0.360 - 0.932i)T^{2} \)
43 \( 1 + (-0.894 + 0.692i)T + (0.249 - 0.968i)T^{2} \)
47 \( 1 + (0.999 - 0.0387i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-1.01 - 1.61i)T + (-0.431 + 0.902i)T^{2} \)
61 \( 1 + (-0.952 - 0.305i)T^{2} \)
67 \( 1 + (1.26 - 0.352i)T + (0.856 - 0.516i)T^{2} \)
71 \( 1 + (0.0581 + 0.998i)T^{2} \)
73 \( 1 + (0.409 - 1.36i)T + (-0.835 - 0.549i)T^{2} \)
79 \( 1 + (0.627 + 0.778i)T^{2} \)
83 \( 1 + (0.675 + 0.984i)T + (-0.360 + 0.932i)T^{2} \)
89 \( 1 + (1.22 + 1.30i)T + (-0.0581 + 0.998i)T^{2} \)
97 \( 1 + (0.00821 - 0.0379i)T + (-0.910 - 0.413i)T^{2} \)
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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

−9.539497843920885787631159214619, −8.877447661688430671686497358651, −7.40241343577763352827365098335, −6.86652880534423094379768851780, −5.98482959445976265037828735202, −5.42155730718731407117491503575, −4.35265420849798662794925784537, −3.97505173273505978699091625348, −2.82857017899644445755033946428, −1.38649200913943490930271219599, 1.42011198199594938235375304263, 2.39778023106155754206202651508, 3.77103533796685721311318607280, 4.49455827080196377260860293992, 5.39384638846166568754251072335, 6.29921488870489601546370521612, 6.62304578449099611450574978380, 7.48925993197758865688332152878, 8.316266318193132000684264730284, 9.398225738990363853792434799934

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