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Properties

Label 2-432-9.4-c3-0-1
Degree $2$
Conductor $432$
Sign $-0.173 - 0.984i$
Analytic cond. $25.4888$
Root an. cond. $5.04864$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)5-s + (−15.5 − 26.8i)7-s + (7.5 + 12.9i)11-s + (18.5 − 32.0i)13-s + 42·17-s + 28·19-s + (−97.5 + 168. i)23-s + (22 + 38.1i)25-s + (55.5 + 96.1i)29-s + (−102.5 + 177. i)31-s + 279·35-s − 166·37-s + (−130.5 + 226. i)41-s + (−21.5 − 37.2i)43-s + (−88.5 − 153. i)47-s + ⋯
L(s)  = 1  + (−0.402 + 0.697i)5-s + (−0.836 − 1.44i)7-s + (0.205 + 0.356i)11-s + (0.394 − 0.683i)13-s + 0.599·17-s + 0.338·19-s + (−0.883 + 1.53i)23-s + (0.175 + 0.304i)25-s + (0.355 + 0.615i)29-s + (−0.593 + 1.02i)31-s + 1.34·35-s − 0.737·37-s + (−0.497 + 0.860i)41-s + (−0.0762 − 0.132i)43-s + (−0.274 − 0.475i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(25.4888\)
Root analytic conductor: \(5.04864\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9414357461\)
\(L(\frac12)\) \(\approx\) \(0.9414357461\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + (4.5 - 7.79i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (15.5 + 26.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.5 + 32.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 42T + 4.91e3T^{2} \)
19 \( 1 - 28T + 6.85e3T^{2} \)
23 \( 1 + (97.5 - 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-55.5 - 96.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (102.5 - 177. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 166T + 5.06e4T^{2} \)
41 \( 1 + (130.5 - 226. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (88.5 + 153. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 114T + 1.48e5T^{2} \)
59 \( 1 + (79.5 - 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (95.5 + 165. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (210.5 - 364. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 156T + 3.57e5T^{2} \)
73 \( 1 - 182T + 3.89e5T^{2} \)
79 \( 1 + (-566.5 - 981. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-541.5 - 937. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-450.5 - 780. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.77917725759148092271318369268, −10.23148323017230714257345455513, −9.410260275127040418797247643904, −7.969441525830167702290528977839, −7.24356115688758323629364233179, −6.55978718274516047111938178397, −5.24115256974848884317836032062, −3.69613215484793512993523963489, −3.31349511894806453452290979204, −1.23253564219819021721158489995, 0.33556561062097868471125577813, 2.13365243459226088686900512136, 3.42685737800402894206051423204, 4.63672822057916849006697735645, 5.83381891554021431946591178676, 6.48104280476927233795848594527, 7.965799518611042591148609586694, 8.774372866397727631277820961578, 9.343681836236453533628441478390, 10.40894919483553110190630612972

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