Properties

Label 2-432-12.11-c1-0-5
Degree $2$
Conductor $432$
Sign $0.5 + 0.866i$
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  − 3i·5-s + 1.73i·7-s + 5.19·11-s − 2·13-s − 6i·17-s − 6.92i·19-s − 4·25-s − 6i·29-s + 5.19i·31-s + 5.19·35-s + 8·37-s + 10.3i·43-s − 10.3·47-s + 4·49-s + 9i·53-s + ⋯
L(s)  = 1  − 1.34i·5-s + 0.654i·7-s + 1.56·11-s − 0.554·13-s − 1.45i·17-s − 1.58i·19-s − 0.800·25-s − 1.11i·29-s + 0.933i·31-s + 0.878·35-s + 1.31·37-s + 1.58i·43-s − 1.51·47-s + 0.571·49-s + 1.23i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22112 - 0.705014i\)
\(L(\frac12)\) \(\approx\) \(1.22112 - 0.705014i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 - 5.19T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 - 3.46iT - 79T^{2} \)
83 \( 1 + 5.19T + 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−11.44573115940155207031950339703, −9.651416395645334013717277303655, −9.238278483263293495798723472733, −8.543582742779613600137910644437, −7.30014750547882064027662363153, −6.24481860918258213460716285256, −5.02174898797827268788226781339, −4.40479283008061550561707377289, −2.68817947782363692601541890126, −1.00369578083096626536781980315, 1.76990784687708791770356965656, 3.44365976333848894796798811627, 4.11412279329988354133145093741, 5.90713250724622261853421486753, 6.66182779471982331097801526214, 7.44301884967625988068297462613, 8.507657893075190312983883693305, 9.797470392729814725165265279139, 10.36377766638315210562276631789, 11.21754663919364921990930186966

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