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Properties

Label 2-432-144.13-c1-0-10
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: $\chi_{432} (253, \cdot )$
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

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

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