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Properties

Label 2-432-48.35-c1-0-30
Degree $2$
Conductor $432$
Sign $-0.999 - 0.0109i$
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  + (0.900 − 1.09i)2-s + (−0.379 − 1.96i)4-s + (−1.29 − 1.29i)5-s − 3.83·7-s + (−2.48 − 1.35i)8-s + (−2.56 + 0.245i)10-s + (1.73 − 1.73i)11-s + (0.145 + 0.145i)13-s + (−3.44 + 4.17i)14-s + (−3.71 + 1.49i)16-s + 2.19i·17-s + (−4.91 + 4.91i)19-s + (−2.04 + 3.02i)20-s + (−0.331 − 3.46i)22-s − 9.44i·23-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)2-s + (−0.189 − 0.981i)4-s + (−0.577 − 0.577i)5-s − 1.44·7-s + (−0.878 − 0.478i)8-s + (−0.812 + 0.0777i)10-s + (0.524 − 0.524i)11-s + (0.0404 + 0.0404i)13-s + (−0.921 + 1.11i)14-s + (−0.928 + 0.372i)16-s + 0.532i·17-s + (−1.12 + 1.12i)19-s + (−0.457 + 0.676i)20-s + (−0.0706 − 0.738i)22-s − 1.97i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.999 - 0.0109i$
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: \((2,\ 432,\ (\ :1/2),\ -0.999 - 0.0109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00556983 + 1.01988i\)
\(L(\frac12)\) \(\approx\) \(0.00556983 + 1.01988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 1.09i)T \)
3 \( 1 \)
good5 \( 1 + (1.29 + 1.29i)T + 5iT^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 + (-1.73 + 1.73i)T - 11iT^{2} \)
13 \( 1 + (-0.145 - 0.145i)T + 13iT^{2} \)
17 \( 1 - 2.19iT - 17T^{2} \)
19 \( 1 + (4.91 - 4.91i)T - 19iT^{2} \)
23 \( 1 + 9.44iT - 23T^{2} \)
29 \( 1 + (-5.42 + 5.42i)T - 29iT^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
37 \( 1 + (-0.955 + 0.955i)T - 37iT^{2} \)
41 \( 1 - 7.05T + 41T^{2} \)
43 \( 1 + (1.66 + 1.66i)T + 43iT^{2} \)
47 \( 1 - 4.20T + 47T^{2} \)
53 \( 1 + (3.96 + 3.96i)T + 53iT^{2} \)
59 \( 1 + (5.64 - 5.64i)T - 59iT^{2} \)
61 \( 1 + (-0.214 - 0.214i)T + 61iT^{2} \)
67 \( 1 + (-3.96 + 3.96i)T - 67iT^{2} \)
71 \( 1 - 0.302iT - 71T^{2} \)
73 \( 1 + 1.89iT - 73T^{2} \)
79 \( 1 - 14.5iT - 79T^{2} \)
83 \( 1 + (6.41 + 6.41i)T + 83iT^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−10.67716707006129085481124196285, −10.04342450828735174907272459018, −9.022212119061001090260198069129, −8.209824853201552851427720233832, −6.39832197038251263627042513486, −6.11005135034887251749300701798, −4.40383816366283028635975826574, −3.81323826556329664243884429175, −2.50784178492340098536822267806, −0.51214893627295827913609067510, 2.91082330938043283732405215123, 3.67623192370714759991739688265, 4.86411518030726363175635404499, 6.19618359628716872765263459832, 6.90881544128501065130818442105, 7.49844285225600501665185224848, 8.902536609264088506695344109892, 9.551385203156121698963642270475, 10.86591158461818632127221415616, 11.80005671872518304267719389497

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