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Properties

Label 2-432-16.5-c1-0-16
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: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797444 - 0.194766i\)
\(L(\frac12)\) \(\approx\) \(0.797444 - 0.194766i\)
\(L(\frac{3}{2})\) 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

−10.96923232993138142267039310582, −10.05947853989099370548723659848, −8.725612414459318461096478391407, −8.435618898811997038395505270500, −7.66206600425150375383599811262, −6.19404001663916620278440957753, −5.53581544173099236344289605262, −4.54118062090613431801561784186, −3.09450593771585160287132825176, −0.60807803871671606116757402802, 1.57487883167063288377037401999, 3.21444850231081704100566165791, 4.00382317047360174179265775828, 5.11666802170108320483790997600, 6.88359491611926614778670133091, 7.60268942770195936152502777062, 8.559774385127405101664353210764, 9.730839694074347164614725071541, 10.46283442620111498296269150660, 11.12510587879720176683228816276

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