Properties

Label 2-432-16.13-c1-0-9
Degree $2$
Conductor $432$
Sign $-0.203 - 0.978i$
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  + (1.37 + 0.340i)2-s + (1.76 + 0.934i)4-s + (−2.75 + 2.75i)5-s + 0.113i·7-s + (2.10 + 1.88i)8-s + (−4.71 + 2.84i)10-s + (−3.69 + 3.69i)11-s + (−1.53 − 1.53i)13-s + (−0.0386 + 0.155i)14-s + (2.25 + 3.30i)16-s + 6.62·17-s + (3.27 + 3.27i)19-s + (−7.43 + 2.29i)20-s + (−6.32 + 3.81i)22-s − 6.20i·23-s + ⋯
L(s)  = 1  + (0.970 + 0.240i)2-s + (0.884 + 0.467i)4-s + (−1.23 + 1.23i)5-s + 0.0428i·7-s + (0.745 + 0.666i)8-s + (−1.49 + 0.898i)10-s + (−1.11 + 1.11i)11-s + (−0.426 − 0.426i)13-s + (−0.0103 + 0.0416i)14-s + (0.563 + 0.826i)16-s + 1.60·17-s + (0.751 + 0.751i)19-s + (−1.66 + 0.512i)20-s + (−1.34 + 0.812i)22-s − 1.29i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.203 - 0.978i$
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: \((2,\ 432,\ (\ :1/2),\ -0.203 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21226 + 1.49090i\)
\(L(\frac12)\) \(\approx\) \(1.21226 + 1.49090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.340i)T \)
3 \( 1 \)
good5 \( 1 + (2.75 - 2.75i)T - 5iT^{2} \)
7 \( 1 - 0.113iT - 7T^{2} \)
11 \( 1 + (3.69 - 3.69i)T - 11iT^{2} \)
13 \( 1 + (1.53 + 1.53i)T + 13iT^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + (-3.27 - 3.27i)T + 19iT^{2} \)
23 \( 1 + 6.20iT - 23T^{2} \)
29 \( 1 + (-1.51 - 1.51i)T + 29iT^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \)
41 \( 1 - 9.18iT - 41T^{2} \)
43 \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \)
47 \( 1 + 3.56T + 47T^{2} \)
53 \( 1 + (-4.08 + 4.08i)T - 53iT^{2} \)
59 \( 1 + (-0.942 + 0.942i)T - 59iT^{2} \)
61 \( 1 + (0.437 + 0.437i)T + 61iT^{2} \)
67 \( 1 + (3.26 + 3.26i)T + 67iT^{2} \)
71 \( 1 + 7.72iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 1.25T + 79T^{2} \)
83 \( 1 + (8.42 + 8.42i)T + 83iT^{2} \)
89 \( 1 + 0.361iT - 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−11.66898495594549112304638301144, −10.53810075391039325245588976388, −10.12827520276198175319080622017, −7.989968588887085286335781767358, −7.64960680219577839825276622477, −6.87045548134411850164782234182, −5.62066604519784882959954525794, −4.54843380097840203385210290280, −3.40651862165833586220761053069, −2.59829679394082883619295069005, 0.942562973207188617340066126406, 3.01909540993960456970441649599, 3.96970074552310840526207984713, 5.11620086351860435309777868915, 5.63262082123175057333636702196, 7.39785012221771984416000209249, 7.86420470772697073933530457300, 9.057081981148840070691499764088, 10.20605524472574194626207074990, 11.34848539812951528017607264628

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