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Properties

Label 2-432-48.11-c1-0-27
Degree $2$
Conductor $432$
Sign $0.844 + 0.536i$
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  + (1.39 − 0.215i)2-s + (1.90 − 0.602i)4-s + (1.33 − 1.33i)5-s + 0.400·7-s + (2.53 − 1.25i)8-s + (1.58 − 2.15i)10-s + (−0.0888 − 0.0888i)11-s + (−3.59 + 3.59i)13-s + (0.559 − 0.0863i)14-s + (3.27 − 2.29i)16-s + 0.898i·17-s + (−3.16 − 3.16i)19-s + (1.74 − 3.35i)20-s + (−0.143 − 0.105i)22-s − 7.09i·23-s + ⋯
L(s)  = 1  + (0.988 − 0.152i)2-s + (0.953 − 0.301i)4-s + (0.598 − 0.598i)5-s + 0.151·7-s + (0.896 − 0.443i)8-s + (0.500 − 0.682i)10-s + (−0.0267 − 0.0267i)11-s + (−0.998 + 0.998i)13-s + (0.149 − 0.0230i)14-s + (0.818 − 0.574i)16-s + 0.217i·17-s + (−0.726 − 0.726i)19-s + (0.390 − 0.750i)20-s + (−0.0305 − 0.0223i)22-s − 1.47i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.844 + 0.536i$
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: \((2,\ 432,\ (\ :1/2),\ 0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60874 - 0.758664i\)
\(L(\frac12)\) \(\approx\) \(2.60874 - 0.758664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.215i)T \)
3 \( 1 \)
good5 \( 1 + (-1.33 + 1.33i)T - 5iT^{2} \)
7 \( 1 - 0.400T + 7T^{2} \)
11 \( 1 + (0.0888 + 0.0888i)T + 11iT^{2} \)
13 \( 1 + (3.59 - 3.59i)T - 13iT^{2} \)
17 \( 1 - 0.898iT - 17T^{2} \)
19 \( 1 + (3.16 + 3.16i)T + 19iT^{2} \)
23 \( 1 + 7.09iT - 23T^{2} \)
29 \( 1 + (-5.95 - 5.95i)T + 29iT^{2} \)
31 \( 1 - 5.25iT - 31T^{2} \)
37 \( 1 + (-0.934 - 0.934i)T + 37iT^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 + (6.81 - 6.81i)T - 43iT^{2} \)
47 \( 1 + 6.60T + 47T^{2} \)
53 \( 1 + (-3.33 + 3.33i)T - 53iT^{2} \)
59 \( 1 + (4.12 + 4.12i)T + 59iT^{2} \)
61 \( 1 + (7.11 - 7.11i)T - 61iT^{2} \)
67 \( 1 + (1.42 + 1.42i)T + 67iT^{2} \)
71 \( 1 - 0.567iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 4.45iT - 79T^{2} \)
83 \( 1 + (6.47 - 6.47i)T - 83iT^{2} \)
89 \( 1 + 6.04T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−11.19572267082799815507205659893, −10.34576922324444196563264697793, −9.376161380452736610001580003192, −8.361264718920180673733318976739, −6.97086225473854969204490630587, −6.32303371592058685720352834445, −4.94676401617772231029293654951, −4.56527259780842976359262822896, −2.88753933770082013880404182325, −1.67519930426694952005988185240, 2.12127054176080006774140680577, 3.14588310813891749134053757343, 4.47379463781167013249199621354, 5.57820365490801458552992448609, 6.30087097138050514004306981355, 7.40838770574420568952752280037, 8.141033918203050740631984930156, 9.802232302900341395189925818831, 10.36049464153221931155502901051, 11.39796511883156337335496287341

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