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Properties

Label 2-432-48.35-c1-0-14
Degree $2$
Conductor $432$
Sign $0.188 + 0.982i$
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  + (−0.205 − 1.39i)2-s + (−1.91 + 0.574i)4-s + (−0.494 − 0.494i)5-s + 4.44·7-s + (1.19 + 2.56i)8-s + (−0.590 + 0.793i)10-s + (0.640 − 0.640i)11-s + (2.56 + 2.56i)13-s + (−0.912 − 6.22i)14-s + (3.34 − 2.20i)16-s − 2.17i·17-s + (−1.65 + 1.65i)19-s + (1.23 + 0.663i)20-s + (−1.02 − 0.764i)22-s − 3.58i·23-s + ⋯
L(s)  = 1  + (−0.145 − 0.989i)2-s + (−0.957 + 0.287i)4-s + (−0.221 − 0.221i)5-s + 1.68·7-s + (0.423 + 0.906i)8-s + (−0.186 + 0.250i)10-s + (0.193 − 0.193i)11-s + (0.710 + 0.710i)13-s + (−0.243 − 1.66i)14-s + (0.835 − 0.550i)16-s − 0.527i·17-s + (−0.380 + 0.380i)19-s + (0.275 + 0.148i)20-s + (−0.219 − 0.163i)22-s − 0.748i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05645 - 0.872880i\)
\(L(\frac12)\) \(\approx\) \(1.05645 - 0.872880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.205 + 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (0.494 + 0.494i)T + 5iT^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 + (-0.640 + 0.640i)T - 11iT^{2} \)
13 \( 1 + (-2.56 - 2.56i)T + 13iT^{2} \)
17 \( 1 + 2.17iT - 17T^{2} \)
19 \( 1 + (1.65 - 1.65i)T - 19iT^{2} \)
23 \( 1 + 3.58iT - 23T^{2} \)
29 \( 1 + (-3.32 + 3.32i)T - 29iT^{2} \)
31 \( 1 + 6.04iT - 31T^{2} \)
37 \( 1 + (3.43 - 3.43i)T - 37iT^{2} \)
41 \( 1 - 9.76T + 41T^{2} \)
43 \( 1 + (-5.78 - 5.78i)T + 43iT^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 + (-8.04 - 8.04i)T + 53iT^{2} \)
59 \( 1 + (9.26 - 9.26i)T - 59iT^{2} \)
61 \( 1 + (2.74 + 2.74i)T + 61iT^{2} \)
67 \( 1 + (-0.758 + 0.758i)T - 67iT^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 - 0.682iT - 73T^{2} \)
79 \( 1 + 6.98iT - 79T^{2} \)
83 \( 1 + (6.26 + 6.26i)T + 83iT^{2} \)
89 \( 1 - 6.59T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−11.15666627861502077740285253085, −10.27679907764247087665311375229, −9.095555771583231251903953055642, −8.370008248597608872586040259868, −7.71267940821492836118405766642, −6.04454936697967296393861382449, −4.64911881307845707439201748384, −4.19233979181789410609799959815, −2.46181264115354486599635666572, −1.20133630083520899344950668885, 1.45275415697418640108236075175, 3.67383277761103317693416226229, 4.83444078620868162632748484908, 5.59611974683998451007104873667, 6.83366951385709879720956042333, 7.76175662311108468209814723408, 8.381995644998604008403064939870, 9.214717331016477251855951948449, 10.59475509972116623645214353510, 11.07272442677869236226705393459

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