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Properties

Label 2-432-27.22-c1-0-12
Degree $2$
Conductor $432$
Sign $0.913 + 0.406i$
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  + (1.70 − 0.327i)3-s + (1.34 − 0.491i)5-s + (−0.111 − 0.629i)7-s + (2.78 − 1.11i)9-s + (−2.56 − 0.933i)11-s + (3.51 + 2.95i)13-s + (2.13 − 1.27i)15-s + (1.37 − 2.38i)17-s + (−2.25 − 3.89i)19-s + (−0.394 − 1.03i)21-s + (−0.868 + 4.92i)23-s + (−2.25 + 1.88i)25-s + (4.37 − 2.80i)27-s + (2.34 − 1.96i)29-s + (−0.510 + 2.89i)31-s + ⋯
L(s)  = 1  + (0.981 − 0.188i)3-s + (0.603 − 0.219i)5-s + (−0.0419 − 0.237i)7-s + (0.928 − 0.371i)9-s + (−0.773 − 0.281i)11-s + (0.976 + 0.819i)13-s + (0.551 − 0.329i)15-s + (0.333 − 0.577i)17-s + (−0.516 − 0.894i)19-s + (−0.0861 − 0.225i)21-s + (−0.181 + 1.02i)23-s + (−0.450 + 0.377i)25-s + (0.841 − 0.539i)27-s + (0.434 − 0.364i)29-s + (−0.0916 + 0.519i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.913 + 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03356 - 0.432171i\)
\(L(\frac12)\) \(\approx\) \(2.03356 - 0.432171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.327i)T \)
good5 \( 1 + (-1.34 + 0.491i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.111 + 0.629i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.56 + 0.933i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.51 - 2.95i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.37 + 2.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.25 + 3.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.868 - 4.92i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.34 + 1.96i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.510 - 2.89i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.60 - 6.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.31 + 6.97i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-9.16 - 3.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.382 + 2.17i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 + (11.1 - 4.07i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.62 - 9.24i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.12 - 4.29i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-6.61 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.02 - 3.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.4 - 8.80i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.98 - 2.50i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (3.94 + 6.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.0 + 5.83i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.98716221394320656771041646831, −10.01265630985826047491456006840, −9.189010340989630242665330920307, −8.496436349327425816857874053665, −7.47044972255809793517801613598, −6.54778298824423613055849954898, −5.32107130426294819517357902032, −4.03198852535739493301595040151, −2.85151183391395682774891514820, −1.53913344465943142354374657330, 1.88746758518588890184772582794, 3.01205657090375101655216110234, 4.16207591739190147117322702163, 5.54454480855555935968684360553, 6.48505918716224281904332573327, 7.899598818486667534919617501526, 8.341255320708223890602852946876, 9.443057324076658926676521692203, 10.34590310347323260407249544667, 10.75395793300610308943984150520

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