Properties

Label 2-432-144.133-c1-0-16
Degree $2$
Conductor $432$
Sign $0.461 + 0.887i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (0.267 − i)5-s + (−2.36 − 1.36i)7-s + (1.99 − 2i)8-s − 1.46i·10-s + (4.23 − 1.13i)11-s + (−3.36 − 0.901i)13-s + (−3.73 − 0.999i)14-s + (1.99 − 3.46i)16-s + 5.73·17-s + (−2.36 + 2.36i)19-s + (−0.535 − 2i)20-s + (5.36 − 3.09i)22-s + (−4.09 + 2.36i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.119 − 0.447i)5-s + (−0.894 − 0.516i)7-s + (0.707 − 0.707i)8-s − 0.462i·10-s + (1.27 − 0.341i)11-s + (−0.933 − 0.250i)13-s + (−0.997 − 0.267i)14-s + (0.499 − 0.866i)16-s + 1.39·17-s + (−0.542 + 0.542i)19-s + (−0.119 − 0.447i)20-s + (1.14 − 0.660i)22-s + (−0.854 + 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.461 + 0.887i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.461 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06371 - 1.25229i\)
\(L(\frac12)\) \(\approx\) \(2.06371 - 1.25229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
good5 \( 1 + (-0.267 + i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.23 + 1.13i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.36 + 0.901i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 + (2.36 - 2.36i)T - 19iT^{2} \)
23 \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.633 - 2.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.267 + 0.464i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.33 - 2.23i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.83 - 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.46 - 7.46i)T + 53iT^{2} \)
59 \( 1 + (-1.96 + 7.33i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3 + 11.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.59 - 1.76i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.366 + 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.20995961869864217951807300257, −9.964895172097931441042223301969, −9.669448470927061634381878978812, −8.116300242509711434264172004038, −6.96470345629069036199717576603, −6.18220274904153142482764237727, −5.15588010148660714451344099554, −3.96631389884197970252020820052, −3.12089302398365724040842611025, −1.32569669481994300730337353410, 2.24934641777783856426148877324, 3.35787598637842151191333601545, 4.44786218821172797067270103508, 5.70472587627970730142466317578, 6.57115600352004861048004696935, 7.18026616249031108967719108142, 8.487561765069370941649830243998, 9.658542602324229213004620003328, 10.40536446229305437427902579785, 11.81014042054159583081021479462

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