Properties

Label 2-432-27.22-c1-0-16
Degree $2$
Conductor $432$
Sign $-0.697 + 0.716i$
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  + (1.43 − 0.972i)3-s + (−3.94 + 1.43i)5-s + (−0.610 − 3.46i)7-s + (1.10 − 2.78i)9-s + (−1.73 − 0.632i)11-s + (−1.78 − 1.49i)13-s + (−4.26 + 5.89i)15-s + (−0.799 + 1.38i)17-s + (−2.31 − 4.00i)19-s + (−4.24 − 4.36i)21-s + (0.308 − 1.74i)23-s + (9.68 − 8.12i)25-s + (−1.12 − 5.07i)27-s + (0.882 − 0.740i)29-s + (−0.322 + 1.83i)31-s + ⋯
L(s)  = 1  + (0.827 − 0.561i)3-s + (−1.76 + 0.642i)5-s + (−0.230 − 1.30i)7-s + (0.369 − 0.929i)9-s + (−0.523 − 0.190i)11-s + (−0.495 − 0.415i)13-s + (−1.10 + 1.52i)15-s + (−0.194 + 0.336i)17-s + (−0.530 − 0.918i)19-s + (−0.925 − 0.953i)21-s + (0.0642 − 0.364i)23-s + (1.93 − 1.62i)25-s + (−0.215 − 0.976i)27-s + (0.163 − 0.137i)29-s + (−0.0579 + 0.328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.697 + 0.716i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331416 - 0.784705i\)
\(L(\frac12)\) \(\approx\) \(0.331416 - 0.784705i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 0.972i)T \)
good5 \( 1 + (3.94 - 1.43i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.610 + 3.46i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.73 + 0.632i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.78 + 1.49i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.799 - 1.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.31 + 4.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.308 + 1.74i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.882 + 0.740i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.322 - 1.83i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.38 - 7.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.98 - 2.50i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.41 - 0.880i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.29 + 7.35i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 8.02T + 53T^{2} \)
59 \( 1 + (-1.15 + 0.418i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.754 + 4.28i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.86 - 4.08i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.871 - 1.50i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.63 + 6.40i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.65 - 7.25i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.71 + 4.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.3 - 4.13i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.79557877469356056446520805710, −10.11816678275971116150299705603, −8.609703746791648187669461473385, −7.966043632483752742586329289602, −7.17760884830961734824831708688, −6.71075335899894517870112158299, −4.54796510800980946442907656515, −3.67889997033067701735128070924, −2.79501008934557966627224325979, −0.47814752169453852311833610508, 2.37953475338763774853710899864, 3.61581817949731594249005974016, 4.51086658221235314216722470368, 5.48269453710103445289964911066, 7.26323968744306459030692389242, 8.000372909957883748053140417040, 8.775701515373259168479354875484, 9.352013818042793189245824602133, 10.60882583893905929496196298005, 11.60669600843594643498094665372

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