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Properties

Label 2-432-9.2-c2-0-8
Degree $2$
Conductor $432$
Sign $0.426 + 0.904i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 2.59i)5-s + (3.17 − 5.49i)7-s + (8.17 + 4.71i)11-s + (−9.84 − 17.0i)13-s + 1.90i·17-s − 4.69·19-s + (8.17 − 4.71i)23-s + (1 − 1.73i)25-s + (2.84 + 1.64i)29-s + (−20.5 − 35.5i)31-s − 32.9i·35-s + 17.3·37-s + (53.5 − 30.9i)41-s + (0.477 − 0.826i)43-s + (−12.2 − 7.05i)47-s + ⋯
L(s)  = 1  + (0.900 − 0.519i)5-s + (0.453 − 0.785i)7-s + (0.743 + 0.429i)11-s + (−0.757 − 1.31i)13-s + 0.112i·17-s − 0.247·19-s + (0.355 − 0.205i)23-s + (0.0400 − 0.0692i)25-s + (0.0982 + 0.0567i)29-s + (−0.662 − 1.14i)31-s − 0.942i·35-s + 0.467·37-s + (1.30 − 0.754i)41-s + (0.0110 − 0.0192i)43-s + (−0.259 − 0.150i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72157 - 1.09118i\)
\(L(\frac12)\) \(\approx\) \(1.72157 - 1.09118i\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-3.17 + 5.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.84 + 17.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 1.90iT - 289T^{2} \)
19 \( 1 + 4.69T + 361T^{2} \)
23 \( 1 + (-8.17 + 4.71i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.84 - 1.64i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (20.5 + 35.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 17.3T + 1.36e3T^{2} \)
41 \( 1 + (-53.5 + 30.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-0.477 + 0.826i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (12.2 + 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 9.53iT - 2.80e3T^{2} \)
59 \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.4 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 85.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0T + 5.32e3T^{2} \)
79 \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (76.1 + 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.63979181261591131805922082710, −9.870828463184935676838240211020, −9.127335030323863647784042746214, −7.966537855803220177262467132523, −7.14042856532831997281312196859, −5.91035756359968016568483141297, −5.02259582802466201235199990554, −3.93986524255537947830211132552, −2.28633801740749565461564413357, −0.916666247310705112355900481396, 1.70241209274272067272022356836, 2.73341082197045377242081591206, 4.30828407233185125079706004098, 5.48923280818245499252296549719, 6.38455822614991526323218099808, 7.21839078051065151387208775637, 8.626913336618908173773457922061, 9.274109848955903287335680171356, 10.09909435099754860459584561233, 11.23697605863208314584637102324

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