Properties

Label 2-432-48.29-c0-0-1
Degree $2$
Conductor $432$
Sign $-0.382 + 0.923i$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)5-s i·7-s + (0.707 − 0.707i)8-s + 1.00i·10-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)5-s i·7-s + (0.707 − 0.707i)8-s + 1.00i·10-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5100651145\)
\(L(\frac12)\) \(\approx\) \(0.5100651145\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−11.11294518251102064731607168112, −10.01909921113000074077632819049, −9.547463512100229922306465248694, −8.353983810850964368888156806829, −7.58874656069789216065624730099, −6.90576961626624250002864950432, −4.85370867393979454556579685243, −4.14131322971884695454873142521, −2.80006230187750664070888616745, −0.906237298384295704270118633130, 2.11170076498663153757087653620, 3.78348720928138463962681262861, 5.20212322128511737537263313873, 6.34078008816720197084693177101, 6.95329677180448468545439805557, 8.122050644798931139573399530374, 8.768093824882733956030420539145, 9.686737008859661650087056706372, 10.67237083024378734677860303803, 11.58301526492345170214753952651

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