Properties

Label 2-1944-72.5-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9439675479\)
\(L(\frac12)\) \(\approx\) \(0.9439675479\)
\(L(1)\) 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

−9.793812155929038254829914243088, −8.880424039839278179118524736826, −7.80607697236205346683616911169, −7.30778215722638949059325731221, −6.79250749747209301839844635956, −6.16565963894889069513453597290, −4.88755013358465858869451796309, −3.96219365418523851713866347892, −3.49264250479041756276115354496, −2.35808603106669681917292818912, 0.59631488487084132584131723922, 1.69851515326486305891240595210, 3.27114415388973101337646966111, 3.90136641072594790008952286025, 4.70688654187092738376079287550, 5.52896532937721665409507982210, 6.18731373860548717780083974756, 7.59314229876548198010508772704, 8.492712020433544476586461693482, 9.044824332087648286267172293979

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