↑

Properties

Label 2-432-9.4-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.800 - 0.598i$
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  + (−1.68 + 2.92i)5-s + (−0.686 − 1.18i)7-s + (−0.5 − 0.866i)11-s + (−2.68 + 4.65i)13-s − 0.372·17-s − 6.37·19-s + (−2.68 + 4.65i)23-s + (−3.18 − 5.51i)25-s + (0.686 + 1.18i)29-s + (0.313 − 0.543i)31-s + 4.62·35-s − 2.74·37-s + (−0.127 + 0.221i)41-s + (4.87 + 8.43i)43-s + (0.686 + 1.18i)47-s + ⋯
L(s)  = 1  + (−0.754 + 1.30i)5-s + (−0.259 − 0.449i)7-s + (−0.150 − 0.261i)11-s + (−0.745 + 1.29i)13-s − 0.0902·17-s − 1.46·19-s + (−0.560 + 0.970i)23-s + (−0.637 − 1.10i)25-s + (0.127 + 0.220i)29-s + (0.0563 − 0.0976i)31-s + 0.782·35-s − 0.451·37-s + (−0.0199 + 0.0345i)41-s + (0.743 + 1.28i)43-s + (0.100 + 0.173i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.800 - 0.598i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196008 + 0.589591i\)
\(L(\frac12)\) \(\approx\) \(0.196008 + 0.589591i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.68 - 2.92i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.686 + 1.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 - 4.65i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.372T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + (2.68 - 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.686 - 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.313 + 0.543i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.74T + 37T^{2} \)
41 \( 1 + (0.127 - 0.221i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 8.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.686 - 1.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.87 + 6.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (0.313 + 0.543i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.68 + 13.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-4.87 - 8.43i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.42072414044366657658905776953, −10.68458115543544981571736239627, −9.905762173403941573076587389261, −8.768348488094884220797014558727, −7.58840994093164300035617255103, −6.97231378291655631167487961381, −6.13013389195367265341969304168, −4.45555077688467848645999613343, −3.58976871816784399937698576073, −2.31146940746556216686206464412, 0.37374736720858796170290176430, 2.41053796528627183459936871492, 4.00574712927983715587259694626, 4.91089788946891995742209696706, 5.86635779328315002377185678761, 7.23206857939337731665914599841, 8.307878095133303794667148467094, 8.696993577635895176696151909565, 9.896986490246650857847752805853, 10.73800109236175697044217697871

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