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Properties

Label 2-432-27.4-c1-0-4
Degree $2$
Conductor $432$
Sign $-0.129 - 0.991i$
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  + (0.631 + 1.61i)3-s + (0.156 − 0.889i)5-s + (−1.02 + 0.860i)7-s + (−2.20 + 2.03i)9-s + (1.03 + 5.84i)11-s + (−0.904 + 0.329i)13-s + (1.53 − 0.308i)15-s + (0.115 − 0.200i)17-s + (0.756 + 1.31i)19-s + (−2.03 − 1.11i)21-s + (3.42 + 2.87i)23-s + (3.93 + 1.43i)25-s + (−4.67 − 2.26i)27-s + (5.21 + 1.89i)29-s + (−7.24 − 6.07i)31-s + ⋯
L(s)  = 1  + (0.364 + 0.931i)3-s + (0.0701 − 0.397i)5-s + (−0.387 + 0.325i)7-s + (−0.734 + 0.678i)9-s + (0.310 + 1.76i)11-s + (−0.250 + 0.0913i)13-s + (0.395 − 0.0795i)15-s + (0.0280 − 0.0486i)17-s + (0.173 + 0.300i)19-s + (−0.444 − 0.242i)21-s + (0.714 + 0.599i)23-s + (0.786 + 0.286i)25-s + (−0.899 − 0.436i)27-s + (0.967 + 0.352i)29-s + (−1.30 − 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921978 + 1.05032i\)
\(L(\frac12)\) \(\approx\) \(0.921978 + 1.05032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.631 - 1.61i)T \)
good5 \( 1 + (-0.156 + 0.889i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.02 - 0.860i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.03 - 5.84i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.904 - 0.329i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.115 + 0.200i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.756 - 1.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.42 - 2.87i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-5.21 - 1.89i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (7.24 + 6.07i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-1.74 + 3.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.06 - 1.84i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.47 + 8.38i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.18 - 4.35i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 + (0.506 - 2.87i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-10.2 + 8.61i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-11.6 + 4.23i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-7.56 + 13.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.83 + 4.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.84 + 2.49i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.91 - 2.15i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-7.28 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.69 + 15.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.31748566575983415578947957653, −10.26368365411191264282432635031, −9.493498307427543608580890650024, −9.044461832972073377314299502747, −7.81385603974541379366232140037, −6.80558028013957160126762787902, −5.36153451671711990237703776877, −4.63931189646785116917144488903, −3.49295004992773951213107931046, −2.09839148368515338752288860184, 0.884882553637367054094836950705, 2.74100828912858479535034959765, 3.53955435479962696403756802361, 5.34574382025852752371843129113, 6.53009944106153686162420709632, 6.95726724781333327057576384356, 8.318001211183013048339294998103, 8.772997482323816158777214887154, 10.04569599182852166652368023650, 11.05799202265537444867959743369

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