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Properties

Label 2-432-144.133-c1-0-10
Degree $2$
Conductor $432$
Sign $0.887 - 0.461i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (−1 + 3.73i)5-s + (0.633 + 0.366i)7-s + (1.99 − 2i)8-s + 5.46i·10-s + (−2.86 + 0.767i)11-s + (6.09 + 1.63i)13-s + (1 + 0.267i)14-s + (1.99 − 3.46i)16-s + 2.26·17-s + (−0.633 + 0.633i)19-s + (2 + 7.46i)20-s + (−3.63 + 2.09i)22-s + (−1.09 + 0.633i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (−0.447 + 1.66i)5-s + (0.239 + 0.138i)7-s + (0.707 − 0.707i)8-s + 1.72i·10-s + (−0.864 + 0.231i)11-s + (1.69 + 0.453i)13-s + (0.267 + 0.0716i)14-s + (0.499 − 0.866i)16-s + 0.550·17-s + (−0.145 + 0.145i)19-s + (0.447 + 1.66i)20-s + (−0.774 + 0.447i)22-s + (−0.228 + 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32911 + 0.569931i\)
\(L(\frac12)\) \(\approx\) \(2.32911 + 0.569931i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
good5 \( 1 + (1 - 3.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 - 0.767i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-6.09 - 1.63i)T + (11.2 + 6.5i)T^{2} \)
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 \( 1 + (0.633 + 2.36i)T + (-25.1 + 14.5i)T^{2} \)
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 \( 1 + (1.23 - 0.330i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.83 + 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.535 - 0.535i)T + 53iT^{2} \)
59 \( 1 + (-1.33 + 4.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.803 - 3i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.23 + 1.40i)T + (58.0 + 33.5i)T^{2} \)
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 \( 1 + (0.366 + 1.36i)T + (-71.8 + 41.5i)T^{2} \)
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

−11.27022710263367631878814166630, −10.65778805159718044664048055568, −9.883297759988596933201536391967, −8.180822846145119444552502177762, −7.29833370781179737730161232097, −6.40201152323026420968147094451, −5.59697326998474436215367664780, −4.05398443062836417607547467566, −3.29064747477102895528235410514, −2.11109531316975509083974371164, 1.35198816614929070096084682030, 3.32024694771716802622464338289, 4.35372151649427583173150479687, 5.26691718582701734904959211320, 5.97260544473672496512982810917, 7.49623789659856252394757604031, 8.276061203960949355067148996991, 8.878922298197713520675730439259, 10.51019844783875139166652876395, 11.26842311943415173326066428385

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