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Properties

Label 2-24e2-36.7-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: $\chi_{576} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6615567767\)
\(L(\frac12)\) \(\approx\) \(0.6615567767\)
\(L(1)\) 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

−10.77714924509815171805591395894, −9.635996116115392236683333563268, −8.580836824936694212061527016454, −8.092713658756460890763013118674, −6.96841031825939149572741285034, −6.27622903084664619547980674391, −5.09654573297028874735518155865, −3.86865233465985760667720547721, −2.57645920488532696922515164157, −0.790781154117031558631967954747, 2.92045987599544459866355883451, 3.28170983337238212184064425614, 4.72922993590790549889667351249, 5.68626648171128731344797541250, 6.69971258034009119277763610877, 7.76986678100632029213242042515, 8.737477136020244760333164536948, 9.658625955781538627348534758090, 10.47284725179511007951499887758, 10.92977523040681339459899696779

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