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Properties

Label 2-432-108.47-c1-0-10
Degree $2$
Conductor $432$
Sign $0.744 + 0.667i$
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  + (−1.59 − 0.672i)3-s + (4.18 − 0.738i)5-s + (1.26 − 1.51i)7-s + (2.09 + 2.14i)9-s + (−0.313 + 1.77i)11-s + (−3.16 − 1.15i)13-s + (−7.17 − 1.63i)15-s + (−1.51 + 0.873i)17-s + (3.62 + 2.09i)19-s + (−3.04 + 1.56i)21-s + (5.40 − 4.53i)23-s + (12.2 − 4.47i)25-s + (−1.90 − 4.83i)27-s + (−3.00 − 8.26i)29-s + (3.17 + 3.78i)31-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)3-s + (1.87 − 0.330i)5-s + (0.479 − 0.571i)7-s + (0.698 + 0.715i)9-s + (−0.0945 + 0.536i)11-s + (−0.878 − 0.319i)13-s + (−1.85 − 0.422i)15-s + (−0.366 + 0.211i)17-s + (0.832 + 0.480i)19-s + (−0.664 + 0.340i)21-s + (1.12 − 0.946i)23-s + (2.45 − 0.894i)25-s + (−0.366 − 0.930i)27-s + (−0.558 − 1.53i)29-s + (0.569 + 0.678i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.744 + 0.667i$
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: \((2,\ 432,\ (\ :1/2),\ 0.744 + 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37015 - 0.523977i\)
\(L(\frac12)\) \(\approx\) \(1.37015 - 0.523977i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 + 0.672i)T \)
good5 \( 1 + (-4.18 + 0.738i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.26 + 1.51i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.313 - 1.77i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (3.16 + 1.15i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.51 - 0.873i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 2.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.40 + 4.53i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.00 + 8.26i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.17 - 3.78i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (-0.864 - 1.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.58 - 4.34i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (9.92 + 1.74i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (2.63 + 2.20i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 0.511iT - 53T^{2} \)
59 \( 1 + (1.75 + 9.98i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.752 - 0.631i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.10 - 11.2i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.03 - 1.78i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.31 + 2.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.02 - 16.5i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.45 + 1.25i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (6.46 + 3.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.250 - 1.42i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.91825463310250156840187423797, −10.06276576918454604546507849398, −9.674767095058676730493014749497, −8.239896739201228920204177951869, −7.07031135884964308580062115511, −6.30544604737671193705083060930, −5.24854801094032674961640886097, −4.72107371591062388529110478085, −2.38491685579519587374679192105, −1.25243975592616411801694547564, 1.61494431747740078319311383829, 2.99725029280773316134367900886, 5.06752127975404585826035010112, 5.32897063722612185762602741739, 6.39010766529543474552921794178, 7.21882671812877477438786849493, 9.058354509156747454514625791428, 9.448738356573156956485406767018, 10.39086214265061525166813269723, 11.15069037382645044141741379689

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