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Properties

Label 2-432-27.4-c1-0-16
Degree $2$
Conductor $432$
Sign $-0.953 + 0.301i$
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  + (0.158 − 1.72i)3-s + (0.536 − 3.04i)5-s + (−3.96 + 3.32i)7-s + (−2.94 − 0.547i)9-s + (−0.729 − 4.13i)11-s + (0.702 − 0.255i)13-s + (−5.16 − 1.40i)15-s + (−0.749 + 1.29i)17-s + (−2.08 − 3.60i)19-s + (5.11 + 7.37i)21-s + (3.23 + 2.71i)23-s + (−4.26 − 1.55i)25-s + (−1.41 + 5.00i)27-s + (−4.93 − 1.79i)29-s + (0.393 + 0.330i)31-s + ⋯
L(s)  = 1  + (0.0916 − 0.995i)3-s + (0.239 − 1.36i)5-s + (−1.49 + 1.25i)7-s + (−0.983 − 0.182i)9-s + (−0.219 − 1.24i)11-s + (0.194 − 0.0708i)13-s + (−1.33 − 0.363i)15-s + (−0.181 + 0.314i)17-s + (−0.477 − 0.827i)19-s + (1.11 + 1.60i)21-s + (0.674 + 0.565i)23-s + (−0.853 − 0.310i)25-s + (−0.271 + 0.962i)27-s + (−0.915 − 0.333i)29-s + (0.0706 + 0.0592i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.953 + 0.301i$
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: \((2,\ 432,\ (\ :1/2),\ -0.953 + 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130802 - 0.847307i\)
\(L(\frac12)\) \(\approx\) \(0.130802 - 0.847307i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.158 + 1.72i)T \)
good5 \( 1 + (-0.536 + 3.04i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (3.96 - 3.32i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.729 + 4.13i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.702 + 0.255i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.749 - 1.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.08 + 3.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 - 2.71i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.93 + 1.79i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.393 - 0.330i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-1.08 + 1.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.58 + 0.939i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.79 + 10.1i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-5.66 + 4.75i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 1.54T + 53T^{2} \)
59 \( 1 + (-1.00 + 5.68i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.80 - 6.55i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.94 + 1.43i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.71 + 4.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.08 - 1.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-15.0 - 5.47i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (2.10 + 0.766i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (1.83 + 3.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.650 - 3.69i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.93170044449677115564849674703, −9.338827840174086183721498965208, −8.936595665735801789968452907485, −8.286353132838842527304878953134, −6.86412638159453762691118041713, −5.87177828818756235040911872077, −5.41296973472438765685694115355, −3.45752049655255805106261639413, −2.27415237447728320849916084996, −0.51013171669342065192485843439, 2.68084257082347660027797403532, 3.57749367610128989595069153951, 4.50386109213067370485854992230, 6.11592792604204315572211792600, 6.83293435914199403873016967591, 7.68041311216587034057251249061, 9.337347168691494050432451624384, 9.889994224877079679027758146380, 10.54535787721231032631938409865, 11.04540345464839244242943744142

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