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Properties

Label 2-432-48.35-c1-0-5
Degree $2$
Conductor $432$
Sign $0.610 - 0.792i$
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  + (−0.932 − 1.06i)2-s + (−0.262 + 1.98i)4-s + (2.82 + 2.82i)5-s − 3.49·7-s + (2.35 − 1.56i)8-s + (0.371 − 5.63i)10-s + (1.04 − 1.04i)11-s + (0.232 + 0.232i)13-s + (3.25 + 3.71i)14-s + (−3.86 − 1.04i)16-s + 6.83i·17-s + (−3.59 + 3.59i)19-s + (−6.34 + 4.85i)20-s + (−2.08 − 0.137i)22-s + 3.02i·23-s + ⋯
L(s)  = 1  + (−0.659 − 0.752i)2-s + (−0.131 + 0.991i)4-s + (1.26 + 1.26i)5-s − 1.31·7-s + (0.832 − 0.554i)8-s + (0.117 − 1.78i)10-s + (0.314 − 0.314i)11-s + (0.0643 + 0.0643i)13-s + (0.869 + 0.992i)14-s + (−0.965 − 0.260i)16-s + 1.65i·17-s + (−0.824 + 0.824i)19-s + (−1.41 + 1.08i)20-s + (−0.444 − 0.0293i)22-s + 0.631i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.610 - 0.792i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.610 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856962 + 0.421711i\)
\(L(\frac12)\) \(\approx\) \(0.856962 + 0.421711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.932 + 1.06i)T \)
3 \( 1 \)
good5 \( 1 + (-2.82 - 2.82i)T + 5iT^{2} \)
7 \( 1 + 3.49T + 7T^{2} \)
11 \( 1 + (-1.04 + 1.04i)T - 11iT^{2} \)
13 \( 1 + (-0.232 - 0.232i)T + 13iT^{2} \)
17 \( 1 - 6.83iT - 17T^{2} \)
19 \( 1 + (3.59 - 3.59i)T - 19iT^{2} \)
23 \( 1 - 3.02iT - 23T^{2} \)
29 \( 1 + (-4.56 + 4.56i)T - 29iT^{2} \)
31 \( 1 + 1.60iT - 31T^{2} \)
37 \( 1 + (-0.0478 + 0.0478i)T - 37iT^{2} \)
41 \( 1 + 5.73T + 41T^{2} \)
43 \( 1 + (-8.10 - 8.10i)T + 43iT^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + (-3.08 - 3.08i)T + 53iT^{2} \)
59 \( 1 + (-2.53 + 2.53i)T - 59iT^{2} \)
61 \( 1 + (5.63 + 5.63i)T + 61iT^{2} \)
67 \( 1 + (4.61 - 4.61i)T - 67iT^{2} \)
71 \( 1 + 5.63iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 4.86iT - 79T^{2} \)
83 \( 1 + (-0.847 - 0.847i)T + 83iT^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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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

−10.86738579616552915250832477836, −10.32960205091160930975726991652, −9.755205440073282196576956241899, −8.913487962581849376932394791676, −7.68336716785149299167692450784, −6.34309126931485142209521778623, −6.17915825294097076858854712451, −3.87149217009457479683989130618, −2.98304918219158601546797137410, −1.85938045451395514467486020238, 0.75477279238512773278046219527, 2.42792775588260558496345737131, 4.60426245235424301223467363634, 5.44504751396264006722279300923, 6.43074867954569100053389938348, 7.09675741612617862742405028704, 8.754631810238676156931250951300, 9.048872526538784969453811144612, 9.834728044737791103476148894557, 10.51006586928738855134169684466

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