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Properties

Label 2-432-9.4-c3-0-10
Degree $2$
Conductor $432$
Sign $0.800 + 0.598i$
Analytic cond. $25.4888$
Root an. cond. $5.04864$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.31 − 4.00i)5-s + (6.05 + 10.4i)7-s + (−5.01 − 8.67i)11-s + (24.2 − 42.0i)13-s − 75.3·17-s + 116.·19-s + (19.0 − 32.9i)23-s + (51.7 + 89.7i)25-s + (−11.3 − 19.5i)29-s + (15.0 − 26.0i)31-s + 56.0·35-s + 130.·37-s + (173. − 300. i)41-s + (13.3 + 23.1i)43-s + (−230. − 399. i)47-s + ⋯
L(s)  = 1  + (0.206 − 0.358i)5-s + (0.327 + 0.566i)7-s + (−0.137 − 0.237i)11-s + (0.518 − 0.897i)13-s − 1.07·17-s + 1.40·19-s + (0.172 − 0.298i)23-s + (0.414 + 0.717i)25-s + (−0.0724 − 0.125i)29-s + (0.0872 − 0.151i)31-s + 0.270·35-s + 0.578·37-s + (0.661 − 1.14i)41-s + (0.0474 + 0.0822i)43-s + (−0.715 − 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(25.4888\)
Root analytic conductor: \(5.04864\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3/2),\ 0.800 + 0.598i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.050547214\)
\(L(\frac12)\) \(\approx\) \(2.050547214\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.31 + 4.00i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-6.05 - 10.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (5.01 + 8.67i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 + 42.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 75.3T + 4.91e3T^{2} \)
19 \( 1 - 116.T + 6.85e3T^{2} \)
23 \( 1 + (-19.0 + 32.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (11.3 + 19.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-15.0 + 26.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 130.T + 5.06e4T^{2} \)
41 \( 1 + (-173. + 300. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-13.3 - 23.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (230. + 399. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 438.T + 1.48e5T^{2} \)
59 \( 1 + (4.18 - 7.24i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 - 71.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-341. + 591. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 - 470.T + 3.89e5T^{2} \)
79 \( 1 + (-243. - 420. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (49.5 + 85.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 8.80T + 7.04e5T^{2} \)
97 \( 1 + (330. + 572. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.76125823322654035513570769360, −9.630070020363806695395834704563, −8.810592037543436346020184860329, −8.044471557765244751056802114139, −6.91404460055179567628906263004, −5.68314630860095399701048891986, −5.05465388374477885316458034626, −3.60162465054209124173443015865, −2.31648076152997183621969163361, −0.804345928770298149899370694724, 1.16532111302845353441154507788, 2.58624797477732001265807063476, 3.97283129420084602998050688102, 4.93812204875895900156632602979, 6.26718557089823110933326563092, 7.06212713793596531588860339997, 8.009881878303925777701405664006, 9.126849573490850659459806100158, 9.900756413251951674899554230490, 10.99199059078609681671320847076

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