Properties

Label 2-432-27.25-c1-0-10
Degree $2$
Conductor $432$
Sign $0.930 + 0.366i$
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  + (−1.01 + 1.40i)3-s + (1.46 − 1.23i)5-s + (3.86 − 1.40i)7-s + (−0.921 − 2.85i)9-s + (−4.34 − 3.64i)11-s + (−0.251 − 1.42i)13-s + (0.228 + 3.31i)15-s + (1.36 + 2.37i)17-s + (2.54 − 4.40i)19-s + (−1.96 + 6.83i)21-s + (4.42 + 1.61i)23-s + (−0.229 + 1.29i)25-s + (4.93 + 1.61i)27-s + (0.768 − 4.35i)29-s + (1.06 + 0.387i)31-s + ⋯
L(s)  = 1  + (−0.588 + 0.808i)3-s + (0.657 − 0.551i)5-s + (1.45 − 0.531i)7-s + (−0.307 − 0.951i)9-s + (−1.31 − 1.10i)11-s + (−0.0696 − 0.394i)13-s + (0.0590 + 0.855i)15-s + (0.332 + 0.575i)17-s + (0.584 − 1.01i)19-s + (−0.429 + 1.49i)21-s + (0.922 + 0.335i)23-s + (−0.0458 + 0.259i)25-s + (0.950 + 0.311i)27-s + (0.142 − 0.809i)29-s + (0.191 + 0.0696i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.930 + 0.366i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34789 - 0.255843i\)
\(L(\frac12)\) \(\approx\) \(1.34789 - 0.255843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.40i)T \)
good5 \( 1 + (-1.46 + 1.23i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.86 + 1.40i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (4.34 + 3.64i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.251 + 1.42i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.36 - 2.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.54 + 4.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.42 - 1.61i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.768 + 4.35i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.06 - 0.387i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.97 - 3.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0493 + 0.280i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.90 - 4.11i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.79 - 2.10i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 8.16T + 53T^{2} \)
59 \( 1 + (10.8 - 9.14i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.76 + 1.00i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.14 - 12.1i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.15 - 5.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.09 + 6.18i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.82 + 16.0i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-1.02 + 1.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.00867 - 0.00727i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.03100002939729280330200683209, −10.39508210699333180156380192227, −9.386926507446668703335263947864, −8.415623306309606394295896806253, −7.59567126371167679489030904355, −5.97595137190361371792965709433, −5.21058491284810973881472651381, −4.61812089283496790395927773824, −3.03983198512966217699432858869, −1.05318347984276992391939747474, 1.70596300952238421001554377673, 2.57214106167724851536069646192, 4.90649589832317892162796446225, 5.34080627511180217667961452574, 6.55780121053067926298778573619, 7.57285463560011606859827679921, 8.112062098792185289852495376414, 9.521752785996130183342143222394, 10.56799429833228053110851815956, 11.14837684353642353300354739450

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