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Properties

Label 2-864-36.31-c0-0-0
Degree $2$
Conductor $864$
Sign $-0.422 - 0.906i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
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)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 0.999i·55-s + (0.866 + 0.5i)59-s + (−0.5 − 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 0.999i·55-s + (0.866 + 0.5i)59-s + (−0.5 − 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :0),\ -0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6115937540\)
\(L(\frac12)\) \(\approx\) \(0.6115937540\)
\(L(1)\) 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 \( 1 \)
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

−10.48767425412967402106391005602, −9.948442271381706015105394871378, −9.008464510124408639028058623800, −8.044508154521647989445174687533, −7.05795311548798955758465472910, −6.57402665427863424024158611834, −5.40865390099123074064642566586, −4.29677699607150867429167970752, −3.14463724932251819374637529893, −2.30605213324246940552615166060, 0.58868079306390712002911967979, 2.63727714629547576615179841532, 3.72169157809914134317815112880, 4.72901007213942848539095926148, 5.67042621352447369701036155002, 6.62713019005439960754788210586, 7.944837254887122468806979503189, 8.072756827968712352794089491626, 9.372335675832358531805963328719, 10.06721434951602460968226377924

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