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Properties

Label 2-432-16.13-c1-0-11
Degree $2$
Conductor $432$
Sign $0.767 - 0.640i$
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  + (−1.36 + 0.379i)2-s + (1.71 − 1.03i)4-s + (2.51 − 2.51i)5-s + 4.16i·7-s + (−1.93 + 2.06i)8-s + (−2.47 + 4.38i)10-s + (−2.68 + 2.68i)11-s + (3.57 + 3.57i)13-s + (−1.58 − 5.67i)14-s + (1.85 − 3.54i)16-s + 1.83·17-s + (0.593 + 0.593i)19-s + (1.70 − 6.90i)20-s + (2.63 − 4.67i)22-s − 3.54i·23-s + ⋯
L(s)  = 1  + (−0.963 + 0.268i)2-s + (0.855 − 0.517i)4-s + (1.12 − 1.12i)5-s + 1.57i·7-s + (−0.685 + 0.728i)8-s + (−0.781 + 1.38i)10-s + (−0.809 + 0.809i)11-s + (0.991 + 0.991i)13-s + (−0.423 − 1.51i)14-s + (0.464 − 0.885i)16-s + 0.445·17-s + (0.136 + 0.136i)19-s + (0.380 − 1.54i)20-s + (0.562 − 0.996i)22-s − 0.739i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02964 + 0.373051i\)
\(L(\frac12)\) \(\approx\) \(1.02964 + 0.373051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.379i)T \)
3 \( 1 \)
good5 \( 1 + (-2.51 + 2.51i)T - 5iT^{2} \)
7 \( 1 - 4.16iT - 7T^{2} \)
11 \( 1 + (2.68 - 2.68i)T - 11iT^{2} \)
13 \( 1 + (-3.57 - 3.57i)T + 13iT^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 + (-0.593 - 0.593i)T + 19iT^{2} \)
23 \( 1 + 3.54iT - 23T^{2} \)
29 \( 1 + (-4.44 - 4.44i)T + 29iT^{2} \)
31 \( 1 + 3.33T + 31T^{2} \)
37 \( 1 + (-4.47 + 4.47i)T - 37iT^{2} \)
41 \( 1 - 2.15iT - 41T^{2} \)
43 \( 1 + (-3.59 + 3.59i)T - 43iT^{2} \)
47 \( 1 - 4.26T + 47T^{2} \)
53 \( 1 + (7.72 - 7.72i)T - 53iT^{2} \)
59 \( 1 + (-2.00 + 2.00i)T - 59iT^{2} \)
61 \( 1 + (7.90 + 7.90i)T + 61iT^{2} \)
67 \( 1 + (-8.67 - 8.67i)T + 67iT^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + 4.88iT - 73T^{2} \)
79 \( 1 - 0.386T + 79T^{2} \)
83 \( 1 + (0.648 + 0.648i)T + 83iT^{2} \)
89 \( 1 + 2.87iT - 89T^{2} \)
97 \( 1 - 9.38T + 97T^{2} \)
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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

−11.06686650321750054608198896714, −10.03810817894778227530879414587, −9.097735575559716300894702885212, −8.934834160425932334759824370493, −7.88008355791063918558766237832, −6.42665367344155963531674800796, −5.70573263238824230184640254093, −4.90753522111091281219627441906, −2.49125109959136922579140712547, −1.58051235817553092700705031422, 1.06157497003582311468599543321, 2.75528764172022754789876285895, 3.61006570011409832365982003600, 5.71552678764882077868898125396, 6.52158531386126731405469320774, 7.50994489935078761115084900618, 8.189105164924458720332595423308, 9.601553006263261300807035996443, 10.28881757988033359026605139510, 10.74243095110043106506485797874

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