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Properties

Label 2-432-9.7-c1-0-0
Degree $2$
Conductor $432$
Sign $0.173 - 0.984i$
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  + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)7-s + (−2.5 + 4.33i)11-s + (2.5 + 4.33i)13-s + 2·17-s + 4·19-s + (0.5 + 0.866i)23-s + (2 − 3.46i)25-s + (−4.5 + 7.79i)29-s + (−0.5 − 0.866i)31-s + 3·35-s − 6·37-s + (1.5 + 2.59i)41-s + (0.5 − 0.866i)43-s + (1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.566 + 0.981i)7-s + (−0.753 + 1.30i)11-s + (0.693 + 1.20i)13-s + 0.485·17-s + 0.917·19-s + (0.104 + 0.180i)23-s + (0.400 − 0.692i)25-s + (−0.835 + 1.44i)29-s + (−0.0898 − 0.155i)31-s + 0.507·35-s − 0.986·37-s + (0.234 + 0.405i)41-s + (0.0762 − 0.132i)43-s + (0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834563 + 0.700282i\)
\(L(\frac12)\) \(\approx\) \(0.834563 + 0.700282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.51177815108204120694486593573, −10.36897861647520783416578977668, −9.394197481561256005738976917049, −8.841304602998409648703408063507, −7.64853519643636250393263359976, −6.73076576302961635812917879929, −5.56049991448361510819767810295, −4.65323200998414081523744658923, −3.27892027375685756721337790678, −1.86510636850193483578702143647, 0.70996835028244278359339028172, 3.07654832083245814081515633381, 3.66384445606605246918677394080, 5.33980905806650582250778550093, 6.17230312203542733600921981594, 7.42987543268824201292823725330, 7.987049205959040255893155906594, 9.178841475932890800747327845910, 10.36696948266601687657928147189, 10.73820080337191516685457593816

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