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Properties

Label 2-1944-72.29-c0-0-8
Degree $2$
Conductor $1944$
Sign $0.0871 + 0.996i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 1.22i)5-s + (−0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s + (−1 − 0.999i)10-s + (0.866 − 0.5i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s − 1.41i·17-s − i·19-s + (−1.22 − 0.707i)20-s + (−1.22 + 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 1.22i)5-s + (−0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s + (−1 − 0.999i)10-s + (0.866 − 0.5i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s − 1.41i·17-s − i·19-s + (−1.22 − 0.707i)20-s + (−1.22 + 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.0871 + 0.996i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.0871 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.791516129\)
\(L(\frac12)\) \(\approx\) \(1.791516129\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
good5 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.160354188644758802538982061915, −8.477591926304119729735823806122, −7.58469156620135781392897183494, −6.65445065339060837349879176631, −5.70583645759446406069060542519, −5.12452840590224191445349673569, −4.33137991996801459195225426831, −3.39402786809487850388162923236, −2.49221584353409145990665859967, −1.00783614577151169499100071847, 1.92480651337774270848425548859, 3.16899701042474434620135318813, 3.94916829315224528102807916422, 4.18776030151012489437413780541, 5.90954167212259057848357839537, 6.32729224529124223321803877214, 7.00961512698481437764824393019, 7.86802138551586987442565991298, 8.316342984026286049345620146311, 9.879927836108669322274161049743

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