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Properties

Label 2-432-27.7-c1-0-5
Degree $2$
Conductor $432$
Sign $0.0581 - 0.998i$
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  + (1.11 + 1.32i)3-s + (0.439 + 2.49i)5-s + (1.79 + 1.50i)7-s + (−0.520 + 2.95i)9-s + (0.745 − 4.22i)11-s + (−0.713 − 0.259i)13-s + (−2.81 + 3.35i)15-s + (−2.46 − 4.26i)17-s + (−3.62 + 6.27i)19-s + 4.06i·21-s + (0.233 − 0.196i)23-s + (−1.32 + 0.482i)25-s + (−4.5 + 2.59i)27-s + (2.91 − 1.06i)29-s + (6.58 − 5.52i)31-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.196 + 1.11i)5-s + (0.679 + 0.570i)7-s + (−0.173 + 0.984i)9-s + (0.224 − 1.27i)11-s + (−0.197 − 0.0719i)13-s + (−0.727 + 0.867i)15-s + (−0.596 − 1.03i)17-s + (−0.831 + 1.44i)19-s + 0.887i·21-s + (0.0487 − 0.0409i)23-s + (−0.265 + 0.0965i)25-s + (−0.866 + 0.499i)27-s + (0.541 − 0.197i)29-s + (1.18 − 0.992i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0581 - 0.998i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0581 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31138 + 1.23722i\)
\(L(\frac12)\) \(\approx\) \(1.31138 + 1.23722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
good5 \( 1 + (-0.439 - 2.49i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.79 - 1.50i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.745 + 4.22i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.713 + 0.259i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.46 + 4.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.62 - 6.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.233 + 0.196i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.91 + 1.06i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.58 + 5.52i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.60 + 1.67i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.283 + 1.60i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.39 - 1.16i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 + (0.950 + 5.39i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.46 - 7.10i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.0393 - 0.0143i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.10 + 3.64i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.54 + 9.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.92 - 2.52i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.41 + 2.33i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (3.96 - 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.570 - 3.23i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.21172016474807995352906472614, −10.45641796749502358262171858263, −9.666267550874947257111711646808, −8.538729994651351442332125213248, −8.031284451456878144149069629616, −6.63751940414403202448072333032, −5.65588704367446864452499614660, −4.43280511971805465369517896679, −3.19876712258709981210185406307, −2.31605451266142563778144995912, 1.20866647377926384729143978350, 2.31892581321857569198198524483, 4.20509667891355279275319543932, 4.89076042415378985126519382349, 6.49442194385314802808385588819, 7.26234350503916298368775522263, 8.350405480499621453395007308568, 8.863759904796456729849386437432, 9.851271912818825381395367196682, 10.99566777652984908393063976174

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