Properties

Label 2-24e2-72.5-c0-0-1
Degree $2$
Conductor $576$
Sign $0.819 + 0.573i$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
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  + (0.866 − 0.5i)3-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−1.5 − 0.866i)33-s + (−1.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.49i)51-s + (0.5 + 0.866i)57-s + (−0.866 + 1.5i)59-s + (0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−1.5 − 0.866i)33-s + (−1.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.49i)51-s + (0.5 + 0.866i)57-s + (−0.866 + 1.5i)59-s + (0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :0),\ 0.819 + 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142121209\)
\(L(\frac12)\) \(\approx\) \(1.142121209\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−10.69917196598393654469637707549, −10.05141441076096763035972071224, −8.681150784649824191435510085034, −8.400872652561530827692562958670, −7.47440832375795933917676830325, −6.33286240630181358207240000623, −5.50040927306322818781316997100, −3.84968304718626400168524045296, −3.07759556465071361272217360998, −1.62958493013945976422657223905, 2.21398352661825444463316834287, 3.09682451146858744606590839211, 4.66303731426903007899028002723, 4.98754688349010874943566305594, 6.83401686200139843374107042774, 7.49225222265461180192907109696, 8.433772542766558385379482395711, 9.429412975801605870662714939127, 9.939509342478836191439210719372, 10.81215224812995381133696050208

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