Properties

Label 2-432-108.47-c1-0-3
Degree $2$
Conductor $432$
Sign $0.333 - 0.942i$
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  + (−0.721 − 1.57i)3-s + (0.976 − 0.172i)5-s + (−3.12 + 3.72i)7-s + (−1.95 + 2.27i)9-s + (−0.961 + 5.45i)11-s + (2.65 + 0.966i)13-s + (−0.976 − 1.41i)15-s + (−0.517 + 0.298i)17-s + (−0.141 − 0.0815i)19-s + (8.12 + 2.23i)21-s + (5.35 − 4.49i)23-s + (−3.77 + 1.37i)25-s + (4.99 + 1.44i)27-s + (0.610 + 1.67i)29-s + (−3.26 − 3.88i)31-s + ⋯
L(s)  = 1  + (−0.416 − 0.909i)3-s + (0.436 − 0.0770i)5-s + (−1.18 + 1.40i)7-s + (−0.652 + 0.757i)9-s + (−0.289 + 1.64i)11-s + (0.736 + 0.268i)13-s + (−0.252 − 0.364i)15-s + (−0.125 + 0.0724i)17-s + (−0.0324 − 0.0187i)19-s + (1.77 + 0.487i)21-s + (1.11 − 0.937i)23-s + (−0.754 + 0.274i)25-s + (0.960 + 0.277i)27-s + (0.113 + 0.311i)29-s + (−0.586 − 0.698i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.333 - 0.942i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710095 + 0.502232i\)
\(L(\frac12)\) \(\approx\) \(0.710095 + 0.502232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.721 + 1.57i)T \)
good5 \( 1 + (-0.976 + 0.172i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (3.12 - 3.72i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.961 - 5.45i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.65 - 0.966i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.517 - 0.298i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.141 + 0.0815i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.35 + 4.49i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.610 - 1.67i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.26 + 3.88i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (-0.726 - 1.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.84 - 10.5i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (6.11 + 1.07i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.37 - 5.35i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 8.40iT - 53T^{2} \)
59 \( 1 + (-1.35 - 7.69i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.16 + 2.65i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.02 - 5.55i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.86 + 3.23i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.05 + 5.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.05 + 11.1i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-9.04 + 3.29i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (1.45 + 0.837i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.34 - 13.2i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.59520024451961799534320872844, −10.39772949478558280440555157758, −9.428129024929133234773671970465, −8.729557581474655747956245748175, −7.45970122869916826670353191860, −6.49796876697456247395739810854, −5.90261204716812143610619936703, −4.80261627386593295737190038850, −2.87099584186690496671963840892, −1.86859400489817547685061838626, 0.57124729815356490714621507208, 3.31737183590611778375281778370, 3.74347023135577383439767728260, 5.33113346118081243616168658161, 6.13588767234530614333084241824, 7.03395617702389711089664440952, 8.461292755096824821525907565689, 9.367704944026746840169316276396, 10.24969045234464734619582382481, 10.77046889013552654056953024862

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