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Properties

Label 2-432-27.4-c1-0-7
Degree $2$
Conductor $432$
Sign $0.743 - 0.669i$
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  + (−1.45 + 0.940i)3-s + (−0.196 + 1.11i)5-s + (2.99 − 2.51i)7-s + (1.23 − 2.73i)9-s + (0.324 + 1.84i)11-s + (0.688 − 0.250i)13-s + (−0.760 − 1.80i)15-s + (−0.944 + 1.63i)17-s + (1.37 + 2.37i)19-s + (−1.99 + 6.47i)21-s + (4.46 + 3.74i)23-s + (3.49 + 1.27i)25-s + (0.782 + 5.13i)27-s + (4.99 + 1.81i)29-s + (−1.02 − 0.861i)31-s + ⋯
L(s)  = 1  + (−0.839 + 0.542i)3-s + (−0.0877 + 0.497i)5-s + (1.13 − 0.949i)7-s + (0.410 − 0.911i)9-s + (0.0979 + 0.555i)11-s + (0.190 − 0.0694i)13-s + (−0.196 − 0.465i)15-s + (−0.229 + 0.396i)17-s + (0.314 + 0.544i)19-s + (−0.434 + 1.41i)21-s + (0.930 + 0.781i)23-s + (0.699 + 0.254i)25-s + (0.150 + 0.988i)27-s + (0.928 + 0.337i)29-s + (−0.184 − 0.154i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.743 - 0.669i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.743 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11968 + 0.429776i\)
\(L(\frac12)\) \(\approx\) \(1.11968 + 0.429776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.940i)T \)
good5 \( 1 + (0.196 - 1.11i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.99 + 2.51i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.324 - 1.84i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.688 + 0.250i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.944 - 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.37 - 2.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.46 - 3.74i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.99 - 1.81i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.02 + 0.861i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.69 - 2.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.68 - 0.614i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.873 + 4.95i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.30 - 1.09i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + (-1.95 + 11.0i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-4.00 + 3.36i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.77 - 0.646i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.09 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.94 + 8.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.6 - 4.22i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (10.9 + 3.99i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.86 - 4.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.0596 + 0.338i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.10047041941858806271154347718, −10.56376032892829431162107675645, −9.763945750588420521850004565172, −8.519202639335075244826865578614, −7.36199570764663661004537464415, −6.68039515523202993660838598976, −5.33621891069289972892059022064, −4.53287962754311816452090860443, −3.48585001159102816200903703958, −1.35491781042669014145476312652, 1.06976758893814698218807919890, 2.55182322370590435894195993184, 4.63368356682946590020536070439, 5.21844234629027587889706846666, 6.24442319537071030236421314043, 7.30141640757458762335848476501, 8.432431483713410146021008407528, 8.919597912690041725009589347154, 10.43903444338181741013050161255, 11.30238050615807118963712614371

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