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Properties

Label 2-432-9.4-c3-0-3
Degree $2$
Conductor $432$
Sign $-0.890 - 0.454i$
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  + (−6.92 + 11.9i)5-s + (15.3 + 26.5i)7-s + (21.9 + 38.0i)11-s + (6.11 − 10.5i)13-s − 76.0·17-s + 44.1·19-s + (39.3 − 68.0i)23-s + (−33.3 − 57.7i)25-s + (−46.3 − 80.3i)29-s + (−71.5 + 123. i)31-s − 425.·35-s − 32.4·37-s + (−167. + 290. i)41-s + (−249. − 431. i)43-s + (140. + 244. i)47-s + ⋯
L(s)  = 1  + (−0.619 + 1.07i)5-s + (0.829 + 1.43i)7-s + (0.601 + 1.04i)11-s + (0.130 − 0.226i)13-s − 1.08·17-s + 0.533·19-s + (0.356 − 0.617i)23-s + (−0.266 − 0.461i)25-s + (−0.297 − 0.514i)29-s + (−0.414 + 0.717i)31-s − 2.05·35-s − 0.144·37-s + (−0.639 + 1.10i)41-s + (−0.883 − 1.53i)43-s + (0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.890 - 0.454i$
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: \((2,\ 432,\ (\ :3/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.438479974\)
\(L(\frac12)\) \(\approx\) \(1.438479974\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.92 - 11.9i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-15.3 - 26.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-21.9 - 38.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.11 + 10.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 76.0T + 4.91e3T^{2} \)
19 \( 1 - 44.1T + 6.85e3T^{2} \)
23 \( 1 + (-39.3 + 68.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (46.3 + 80.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (71.5 - 123. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 32.4T + 5.06e4T^{2} \)
41 \( 1 + (167. - 290. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (249. + 431. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-140. - 244. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 628.T + 1.48e5T^{2} \)
59 \( 1 + (252. - 437. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (185. + 322. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (81.3 - 140. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 433.T + 3.57e5T^{2} \)
73 \( 1 + 629.T + 3.89e5T^{2} \)
79 \( 1 + (-86.3 - 149. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-87.4 - 151. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 336.T + 7.04e5T^{2} \)
97 \( 1 + (42.1 + 73.0i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.28047410035701244660470409146, −10.35779456018455401185458771613, −9.179089866010545083259512908396, −8.441318957885998607343805793167, −7.34259974804988356250551071267, −6.59050428146243355153276393983, −5.36590699786218827201142200309, −4.29472883958698800722248547966, −2.92258370283736793171487750836, −1.88521954042354041232958766442, 0.49135726252674069610118451686, 1.46057386210535738927917856940, 3.63871941325803115186987531067, 4.35888272961016239601942745539, 5.31952230396786253254015676131, 6.76543356199108393010424069278, 7.67978664136904650093866855614, 8.519927193899569603015086230562, 9.225654588517308564091878127861, 10.56146063106554445987496295209

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