Properties

Label 2-432-144.61-c1-0-5
Degree $2$
Conductor $432$
Sign $0.953 + 0.300i$
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s + (3.73 + i)5-s + (−0.633 − 0.366i)7-s + (2 + 1.99i)8-s − 5.46i·10-s + (0.767 + 2.86i)11-s + (−1.63 + 6.09i)13-s + (−0.267 + i)14-s + (1.99 − 3.46i)16-s + 2.26·17-s + (−0.633 − 0.633i)19-s + (−7.46 + 2i)20-s + (3.63 − 2.09i)22-s + (1.09 − 0.633i)23-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s + (1.66 + 0.447i)5-s + (−0.239 − 0.138i)7-s + (0.707 + 0.707i)8-s − 1.72i·10-s + (0.231 + 0.864i)11-s + (−0.453 + 1.69i)13-s + (−0.0716 + 0.267i)14-s + (0.499 − 0.866i)16-s + 0.550·17-s + (−0.145 − 0.145i)19-s + (−1.66 + 0.447i)20-s + (0.774 − 0.447i)22-s + (0.228 − 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41654 - 0.218027i\)
\(L(\frac12)\) \(\approx\) \(1.41654 - 0.218027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 \)
good5 \( 1 + (-3.73 - i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.767 - 2.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.63 - 6.09i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + (0.633 + 0.633i)T + 19iT^{2} \)
23 \( 1 + (-1.09 + 0.633i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.36 + 0.633i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.73 + 6.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.330 - 1.23i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.83 + 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.535 + 0.535i)T - 53iT^{2} \)
59 \( 1 + (4.96 + 1.33i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3 - 0.803i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.40 + 5.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.36 + 0.366i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98153410339503650422387458448, −10.01044209776260735413163826349, −9.592417303227873936231112789758, −8.926259286453163231788248956705, −7.35190037863788649384359848082, −6.45105419250351055909150362279, −5.20553773019301408903148037016, −4.06649813398219428227637708107, −2.50280141676677704915967407565, −1.74033945150501020624702714326, 1.14740151092014100777846884239, 3.04285522996453146049582391337, 4.92376946712246516549545537689, 5.72327659372728644057569859643, 6.20240246848151539746791729411, 7.52466291892398585419045296132, 8.557824465782328692079677271442, 9.281251709363666912986230880812, 10.09051627196268123266074158125, 10.70963938639085829304255753234

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