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Properties

Label 2-1584-99.76-c0-0-1
Degree $2$
Conductor $1584$
Sign $0.766 - 0.642i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
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)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 0.999·15-s + (−0.5 + 0.866i)23-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)33-s + 2·37-s + (0.499 + 0.866i)45-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s − 53-s + 0.999·55-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 0.999·15-s + (−0.5 + 0.866i)23-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)33-s + 2·37-s + (0.499 + 0.866i)45-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s − 53-s + 0.999·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.435105046\)
\(L(\frac12)\) \(\approx\) \(1.435105046\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.655082177404250088908352705671, −9.152367502032170571825487790828, −8.208878067122560617452727879756, −7.57809636729813266891741063091, −6.29408666607692165206353021927, −5.45370370567122795889045044044, −4.58150081506149594909596163789, −4.00563733148063794800916949133, −2.71554235261635030335787003885, −1.61529422882588205942478152176, 1.28337414432604416465908669943, 2.59670462080285068610615492056, 3.15011994749559398998350550237, 4.38318827217743050595204559298, 5.83242945228106489959036104831, 6.40916663564188583176283563927, 6.96284390107418333763037588438, 8.016154598522925904965645307102, 8.580989296159898624684766123037, 9.450063770868384759010766385784

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