Properties

Label 2-432-108.47-c1-0-5
Degree $2$
Conductor $432$
Sign $-0.0117 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.340 + 1.69i)3-s + (1.68 − 0.296i)5-s + (−0.981 + 1.16i)7-s + (−2.76 + 1.15i)9-s + (−0.602 + 3.41i)11-s + (2.74 + 0.999i)13-s + (1.07 + 2.75i)15-s + (0.812 − 0.469i)17-s + (1.51 + 0.875i)19-s + (−2.32 − 1.26i)21-s + (0.0294 − 0.0247i)23-s + (−1.95 + 0.712i)25-s + (−2.90 − 4.30i)27-s + (−1.84 − 5.07i)29-s + (4.26 + 5.08i)31-s + ⋯
L(s)  = 1  + (0.196 + 0.980i)3-s + (0.752 − 0.132i)5-s + (−0.370 + 0.442i)7-s + (−0.922 + 0.385i)9-s + (−0.181 + 1.03i)11-s + (0.761 + 0.277i)13-s + (0.277 + 0.711i)15-s + (0.197 − 0.113i)17-s + (0.348 + 0.200i)19-s + (−0.506 − 0.276i)21-s + (0.00614 − 0.00515i)23-s + (−0.391 + 0.142i)25-s + (−0.559 − 0.828i)27-s + (−0.343 − 0.942i)29-s + (0.766 + 0.913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.0117 - 0.999i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.0117 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07405 + 1.08678i\)
\(L(\frac12)\) \(\approx\) \(1.07405 + 1.08678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.340 - 1.69i)T \)
good5 \( 1 + (-1.68 + 0.296i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.981 - 1.16i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.602 - 3.41i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.74 - 0.999i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.812 + 0.469i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.51 - 0.875i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0294 + 0.0247i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.84 + 5.07i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.26 - 5.08i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.09 - 5.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.60 + 4.41i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (8.03 + 1.41i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.29 - 5.27i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.6iT - 53T^{2} \)
59 \( 1 + (2.57 + 14.6i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.6 - 8.91i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-4.52 + 12.4i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.81 + 3.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.02 - 3.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.29 + 11.7i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.42 + 1.24i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-7.61 - 4.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.737 + 4.18i)T + (-91.1 - 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.28908939961646953157269397320, −10.13703555977666565724098008644, −9.723882324122620809929116615855, −8.917860603491262067020834411135, −7.917636457851485282128444404368, −6.48711575523707369612578564900, −5.56989198703419491509766583011, −4.62645460754591494957417153871, −3.39417889074932330824459752106, −2.08559527929647455482875403895, 1.01207457729414354496858561296, 2.56604961321847992584571383434, 3.67910526610092153042000511153, 5.59699931135718728824954697218, 6.16790224496807567965676103028, 7.17154910347173566727300307323, 8.161603043595235570833808506932, 8.985518343316502180905927186764, 10.04197933587720385374309320152, 10.98992933537931609246356209430

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