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Properties

Label 2-432-108.83-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.354 - 0.934i$
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  + (0.536 + 1.64i)3-s + (−1.32 − 1.57i)5-s + (1.34 + 3.70i)7-s + (−2.42 + 1.76i)9-s + (−1.12 − 0.948i)11-s + (0.716 + 4.06i)13-s + (1.88 − 3.02i)15-s + (−4.21 + 2.43i)17-s + (−0.0640 − 0.0369i)19-s + (−5.37 + 4.20i)21-s + (7.06 + 2.56i)23-s + (0.134 − 0.760i)25-s + (−4.20 − 3.04i)27-s + (4.84 + 0.855i)29-s + (−2.86 + 7.87i)31-s + ⋯
L(s)  = 1  + (0.309 + 0.950i)3-s + (−0.591 − 0.704i)5-s + (0.509 + 1.39i)7-s + (−0.808 + 0.588i)9-s + (−0.340 − 0.285i)11-s + (0.198 + 1.12i)13-s + (0.486 − 0.780i)15-s + (−1.02 + 0.590i)17-s + (−0.0146 − 0.00848i)19-s + (−1.17 + 0.917i)21-s + (1.47 + 0.535i)23-s + (0.0268 − 0.152i)25-s + (−0.810 − 0.586i)27-s + (0.900 + 0.158i)29-s + (−0.514 + 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.354 - 0.934i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.354 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703424 + 1.01929i\)
\(L(\frac12)\) \(\approx\) \(0.703424 + 1.01929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.536 - 1.64i)T \)
good5 \( 1 + (1.32 + 1.57i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.34 - 3.70i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.12 + 0.948i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.716 - 4.06i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (4.21 - 2.43i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0640 + 0.0369i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.06 - 2.56i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-4.84 - 0.855i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.86 - 7.87i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.58 + 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.47 - 1.49i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.636 - 0.758i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.18 + 2.25i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 7.92iT - 53T^{2} \)
59 \( 1 + (0.418 - 0.351i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-11.8 + 4.29i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-11.0 + 1.94i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.59 - 6.23i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.52 + 7.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.35 + 0.414i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.677 + 3.84i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-14.2 - 8.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.1 - 8.50i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.40682119804772732258301719842, −10.65457818638724368758395169325, −9.307993177014463720269711965654, −8.659042819997476779622497069836, −8.367066262334014061680830376811, −6.72165316497726017269434984794, −5.29945536999077463956039227649, −4.77541974299133300514856400001, −3.55672018622026616815989436589, −2.14715426412568448442225737001, 0.77342552827768107040928647407, 2.60069571026361890983468301820, 3.71056141263203729969192458789, 5.02719216909894806049125337610, 6.59370572038613140181815472705, 7.26103394681018925334538075735, 7.83432437610930612653090106958, 8.795718046581064687440183230480, 10.23441825596549829214204938670, 10.97898579465777696128871745826

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