Properties

Label 2-432-27.7-c1-0-12
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: $\chi_{432} (385, \cdot )$
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

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

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