Properties

Label 2-432-36.23-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.173 - 0.984i$
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  + (−1.5 − 0.866i)5-s + (−2.59 + 1.5i)7-s + (2.59 + 4.5i)11-s + (−0.5 + 0.866i)13-s + 3.46i·17-s + 6i·19-s + (−2.59 + 4.5i)23-s + (−1 − 1.73i)25-s + (7.5 − 4.33i)29-s + (−2.59 − 1.5i)31-s + 5.19·35-s − 4·37-s + (−4.5 − 2.59i)41-s + (−2.59 + 1.5i)43-s + (−2.59 − 4.5i)47-s + ⋯
L(s)  = 1  + (−0.670 − 0.387i)5-s + (−0.981 + 0.566i)7-s + (0.783 + 1.35i)11-s + (−0.138 + 0.240i)13-s + 0.840i·17-s + 1.37i·19-s + (−0.541 + 0.938i)23-s + (−0.200 − 0.346i)25-s + (1.39 − 0.804i)29-s + (−0.466 − 0.269i)31-s + 0.878·35-s − 0.657·37-s + (−0.702 − 0.405i)41-s + (−0.396 + 0.228i)43-s + (−0.378 − 0.656i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531321 + 0.633204i\)
\(L(\frac12)\) \(\approx\) \(0.531321 + 0.633204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (2.59 - 4.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.5 + 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.59 - 1.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-12.9 + 7.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.85328873377763675528243148773, −10.21736546923397834118685118461, −9.715881578344310113458260278117, −8.688157965834828313886385117844, −7.78279468507188299007012152249, −6.70689262348291192495058544204, −5.82548973559547275081009165802, −4.40724668051379413981439624924, −3.58536186051464766832966500005, −1.88561405254783143931675338137, 0.52363058145015849694816103396, 2.98602229952351743843644484765, 3.69337743861735498078780936030, 5.04736710289279896835066056162, 6.55107730880931695185568681008, 6.91190831286809726056925378783, 8.210957469752214655922877825642, 9.065481023467991825086194440324, 10.06516401614630212499355830310, 11.00236751723428702929993568910

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