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Properties

Label 2-432-27.25-c1-0-16
Degree $2$
Conductor $432$
Sign $-0.893 + 0.448i$
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  + (0.592 − 1.62i)3-s + (−0.673 + 0.565i)5-s + (−3.31 + 1.20i)7-s + (−2.29 − 1.92i)9-s + (−2.73 − 2.29i)11-s + (−0.641 − 3.63i)13-s + (0.520 + 1.43i)15-s + (−3.12 − 5.41i)17-s + (2.08 − 3.61i)19-s + 6.11i·21-s + (1.93 + 0.705i)23-s + (−0.733 + 4.16i)25-s + (−4.5 + 2.59i)27-s + (0.0282 − 0.160i)29-s + (1.53 + 0.560i)31-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)3-s + (−0.301 + 0.252i)5-s + (−1.25 + 0.456i)7-s + (−0.766 − 0.642i)9-s + (−0.825 − 0.692i)11-s + (−0.177 − 1.00i)13-s + (0.134 + 0.369i)15-s + (−0.757 − 1.31i)17-s + (0.478 − 0.828i)19-s + 1.33i·21-s + (0.404 + 0.147i)23-s + (−0.146 + 0.832i)25-s + (−0.866 + 0.499i)27-s + (0.00524 − 0.0297i)29-s + (0.276 + 0.100i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.893 + 0.448i$
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: \((2,\ 432,\ (\ :1/2),\ -0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165294 - 0.697433i\)
\(L(\frac12)\) \(\approx\) \(0.165294 - 0.697433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 + 1.62i)T \)
good5 \( 1 + (0.673 - 0.565i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (3.31 - 1.20i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.73 + 2.29i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.641 + 3.63i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.12 + 5.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.08 + 3.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.93 - 0.705i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.0282 + 0.160i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.53 - 0.560i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-3.85 - 6.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.33 + 7.58i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-8.29 - 6.95i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (6.02 - 2.19i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 0.716T + 53T^{2} \)
59 \( 1 + (-5.35 + 4.49i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.19 + 0.433i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.624 + 3.54i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.76 + 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.16 + 2.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.14 + 6.51i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.773 + 4.38i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (4.62 - 8.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.64 + 7.25i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.96074005977437909093580808920, −9.659867316646388682591360972806, −8.955658758979343981397428374837, −7.86861667815084623898597447450, −7.10876600939628992538908572327, −6.19680899576363822126119983740, −5.18434617033175072112680681030, −3.12597284972154126003472924133, −2.78424417039134136397577512886, −0.40895929664723854727599037438, 2.43684767025811834944237320512, 3.81190728506255511307637807657, 4.44110474654710692767280352875, 5.79749695874518280617044680362, 6.90827521334202489768392255435, 8.028363517842984295116415946072, 8.958009426861310863486478879939, 9.887536515313805043736013684857, 10.33130671066028164669181054473, 11.37679057025669110736747643222

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