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Properties

Label 2-24e2-36.31-c0-0-1
Degree $2$
Conductor $576$
Sign $0.642 + 0.766i$
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  − i·3-s + (−0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − 9-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999i·35-s + (−0.866 − 0.5i)39-s + ⋯
L(s)  = 1  − i·3-s + (−0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − 9-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999i·35-s + (−0.866 − 0.5i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9148160124\)
\(L(\frac12)\) \(\approx\) \(0.9148160124\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−11.18371574850783769859047347762, −10.17388558099597290743556008949, −8.682819294037206968330126167999, −8.076201584535862442229125612919, −7.23596545693276334720937363450, −6.50045814268115577020730499212, −5.48408055588524979420906239569, −3.96662470882946593728461509562, −2.92860072721744225346651211267, −1.35958915097265115570944635848, 1.86744317295586051491063949714, 3.73367026235311878184344605650, 4.45822381830485025157439118835, 5.24386387813607860118779185619, 6.37043589480366613509618615921, 7.79253871594579130471709185130, 8.703878767882091210837063038479, 9.106680844869719325982788247661, 10.10973466519298700024968334687, 11.19236130016779560948715829983

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