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Properties

Label 2-432-48.11-c1-0-28
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: $\chi_{432} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.0474 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50975 - 1.58317i\)
\(L(\frac12)\) \(\approx\) \(1.50975 - 1.58317i\)
\(L(\frac{3}{2})\) 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

−11.35088461192689164240729794290, −9.902506660441623794106071721155, −9.588362552154768531548720768782, −8.214189147475873790012285606024, −7.03380155173665748101927137824, −5.75597595569647808043659128958, −5.20962825616350126740349416655, −3.89740029192567118699910793969, −2.77146610877424905348936233534, −1.24696169349403057535259340053, 2.24863156910838076393562387526, 3.56535966802823605130460559043, 4.66440778287928773512853686055, 5.78186786125813333166481956344, 6.57331180102311872266585032082, 7.50242076798058934635399716690, 8.479339522630622023797373712792, 9.453666262579045118457184278688, 10.72019723938759683403872042001, 11.41553278497424165952060157158

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