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Properties

Label 2-432-16.13-c1-0-18
Degree $2$
Conductor $432$
Sign $0.887 - 0.460i$
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  + (0.305 + 1.38i)2-s + (−1.81 + 0.844i)4-s + (1.45 − 1.45i)5-s − 2.37i·7-s + (−1.72 − 2.24i)8-s + (2.45 + 1.56i)10-s + (3.09 − 3.09i)11-s + (3.80 + 3.80i)13-s + (3.28 − 0.727i)14-s + (2.57 − 3.06i)16-s − 0.668·17-s + (2.02 + 2.02i)19-s + (−1.40 + 3.86i)20-s + (5.22 + 3.32i)22-s − 4.17i·23-s + ⋯
L(s)  = 1  + (0.216 + 0.976i)2-s + (−0.906 + 0.422i)4-s + (0.650 − 0.650i)5-s − 0.898i·7-s + (−0.608 − 0.793i)8-s + (0.776 + 0.494i)10-s + (0.933 − 0.933i)11-s + (1.05 + 1.05i)13-s + (0.877 − 0.194i)14-s + (0.643 − 0.765i)16-s − 0.162·17-s + (0.464 + 0.464i)19-s + (−0.315 + 0.864i)20-s + (1.11 + 0.709i)22-s − 0.869i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.887 - 0.460i$
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: \((2,\ 432,\ (\ :1/2),\ 0.887 - 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58787 + 0.387819i\)
\(L(\frac12)\) \(\approx\) \(1.58787 + 0.387819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.305 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (-1.45 + 1.45i)T - 5iT^{2} \)
7 \( 1 + 2.37iT - 7T^{2} \)
11 \( 1 + (-3.09 + 3.09i)T - 11iT^{2} \)
13 \( 1 + (-3.80 - 3.80i)T + 13iT^{2} \)
17 \( 1 + 0.668T + 17T^{2} \)
19 \( 1 + (-2.02 - 2.02i)T + 19iT^{2} \)
23 \( 1 + 4.17iT - 23T^{2} \)
29 \( 1 + (5.21 + 5.21i)T + 29iT^{2} \)
31 \( 1 - 0.437T + 31T^{2} \)
37 \( 1 + (7.02 - 7.02i)T - 37iT^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 + (-4.50 + 4.50i)T - 43iT^{2} \)
47 \( 1 - 9.31T + 47T^{2} \)
53 \( 1 + (-5.33 + 5.33i)T - 53iT^{2} \)
59 \( 1 + (10.0 - 10.0i)T - 59iT^{2} \)
61 \( 1 + (3.47 + 3.47i)T + 61iT^{2} \)
67 \( 1 + (7.26 + 7.26i)T + 67iT^{2} \)
71 \( 1 + 3.51iT - 71T^{2} \)
73 \( 1 - 15.2iT - 73T^{2} \)
79 \( 1 + 9.00T + 79T^{2} \)
83 \( 1 + (3.40 + 3.40i)T + 83iT^{2} \)
89 \( 1 + 2.38iT - 89T^{2} \)
97 \( 1 + 1.45T + 97T^{2} \)
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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

−11.28732604498724785163509086432, −10.08934053676727898788867837761, −9.050234700933904490703993520759, −8.638911456173647980028761175281, −7.39341952356490594825216494652, −6.40316467833783030047983510394, −5.74914568481226445775353050593, −4.42016675199095381010898087728, −3.65270398833177978499841277908, −1.19834447667884066140718790914, 1.64111426809420509323987585198, 2.80461689518320605856497361391, 3.90287511140427008776551193037, 5.37768085913855391579010812198, 6.03186099283831485327263705508, 7.36216376546413751687044066149, 8.891301967532602892014982334084, 9.289778373205943397797992521584, 10.39043346773415260483936843802, 10.99008416335393760589617983076

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