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Properties

Label 2-432-48.11-c1-0-12
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: $\chi_{432} (107, \cdot )$
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

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

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