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Properties

Label 2-432-48.35-c1-0-12
Degree $2$
Conductor $432$
Sign $0.0474 + 0.998i$
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  + (−1.10 − 0.884i)2-s + (0.436 + 1.95i)4-s + (−0.800 − 0.800i)5-s + 0.180·7-s + (1.24 − 2.54i)8-s + (0.175 + 1.59i)10-s + (0.626 − 0.626i)11-s + (1.46 + 1.46i)13-s + (−0.199 − 0.159i)14-s + (−3.61 + 1.70i)16-s − 4.61i·17-s + (3.52 − 3.52i)19-s + (1.21 − 1.91i)20-s + (−1.24 + 0.137i)22-s + 2.90i·23-s + ⋯
L(s)  = 1  + (−0.780 − 0.625i)2-s + (0.218 + 0.975i)4-s + (−0.357 − 0.357i)5-s + 0.0683·7-s + (0.439 − 0.898i)8-s + (0.0555 + 0.503i)10-s + (0.189 − 0.189i)11-s + (0.407 + 0.407i)13-s + (−0.0533 − 0.0427i)14-s + (−0.904 + 0.426i)16-s − 1.11i·17-s + (0.809 − 0.809i)19-s + (0.271 − 0.427i)20-s + (−0.265 + 0.0293i)22-s + 0.605i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0474 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0474 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0474 + 0.998i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0474 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630420 - 0.601184i\)
\(L(\frac12)\) \(\approx\) \(0.630420 - 0.601184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.884i)T \)
3 \( 1 \)
good5 \( 1 + (0.800 + 0.800i)T + 5iT^{2} \)
7 \( 1 - 0.180T + 7T^{2} \)
11 \( 1 + (-0.626 + 0.626i)T - 11iT^{2} \)
13 \( 1 + (-1.46 - 1.46i)T + 13iT^{2} \)
17 \( 1 + 4.61iT - 17T^{2} \)
19 \( 1 + (-3.52 + 3.52i)T - 19iT^{2} \)
23 \( 1 - 2.90iT - 23T^{2} \)
29 \( 1 + (-4.85 + 4.85i)T - 29iT^{2} \)
31 \( 1 + 1.16iT - 31T^{2} \)
37 \( 1 + (-6.89 + 6.89i)T - 37iT^{2} \)
41 \( 1 + 4.74T + 41T^{2} \)
43 \( 1 + (2.46 + 2.46i)T + 43iT^{2} \)
47 \( 1 - 4.17T + 47T^{2} \)
53 \( 1 + (5.71 + 5.71i)T + 53iT^{2} \)
59 \( 1 + (5.43 - 5.43i)T - 59iT^{2} \)
61 \( 1 + (-4.51 - 4.51i)T + 61iT^{2} \)
67 \( 1 + (-10.3 + 10.3i)T - 67iT^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \)
89 \( 1 - 2.64T + 89T^{2} \)
97 \( 1 + 6.85T + 97T^{2} \)
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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

−11.12829936653937002462748522965, −9.863409801630763402456446812726, −9.249569539840247080982365938056, −8.332798313573265289434863553977, −7.49894349167188648150154337091, −6.49325960615959220839014658834, −4.90267465951832132988523969916, −3.77928898239160649562153456474, −2.51100448702881242549115832674, −0.807210823308297313707765278575, 1.44763904890839545901369861904, 3.28086063156772420797635599504, 4.79230749154090428063260947238, 5.98284757170374579511517949534, 6.78201001757356178608900348927, 7.86045478601589335510435833330, 8.423575562554990942087555820710, 9.549527596258171714051562089440, 10.38697972680089692679694796814, 11.08391653824910099243836228508

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