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Properties

Label 2-432-36.11-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.342 - 0.939i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)5-s + (−1.5 − 0.866i)7-s + (−1.5 + 2.59i)11-s + (2.5 + 4.33i)13-s + 6.92i·17-s + 3.46i·19-s + (−4.5 − 7.79i)23-s + (−1 + 1.73i)25-s + (−1.5 − 0.866i)29-s + (−4.5 + 2.59i)31-s + 3·35-s + 2·37-s + (4.5 − 2.59i)41-s + (4.5 + 2.59i)43-s + (−1.5 + 2.59i)47-s + ⋯
L(s)  = 1  + (−0.670 + 0.387i)5-s + (−0.566 − 0.327i)7-s + (−0.452 + 0.783i)11-s + (0.693 + 1.20i)13-s + 1.68i·17-s + 0.794i·19-s + (−0.938 − 1.62i)23-s + (−0.200 + 0.346i)25-s + (−0.278 − 0.160i)29-s + (−0.808 + 0.466i)31-s + 0.507·35-s + 0.328·37-s + (0.702 − 0.405i)41-s + (0.686 + 0.396i)43-s + (−0.218 + 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463452 + 0.661879i\)
\(L(\frac12)\) \(\approx\) \(0.463452 + 0.661879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (7.5 + 4.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.5 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.30666223470059285580297598618, −10.56410182976454161878133929011, −9.770132792621821597984250814948, −8.603698370434085266204053662306, −7.75339736374419552933253366361, −6.75892452099719041101830918755, −5.96855201868062485240538828118, −4.28492090065725344306778087852, −3.68434356292360931445639533549, −1.97753131598309101828180766124, 0.50416412224080305732476102584, 2.79531521174003500966884648026, 3.77370142722726638115860021834, 5.21610563014850485601102771283, 5.96937158313213942312223206075, 7.37297901985174399572602851810, 8.074955773744659062333654214197, 9.073722340683754912567250884987, 9.869661786687927129368596611314, 11.11891800362927334959354332949

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