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Properties

Label 2-432-144.85-c1-0-6
Degree $2$
Conductor $432$
Sign $-0.300 - 0.953i$
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 + (−1 + 0.267i)5-s + (2.36 − 1.36i)7-s + (2 − 1.99i)8-s − 1.46i·10-s + (−1.13 + 4.23i)11-s + (0.901 + 3.36i)13-s + (0.999 + 3.73i)14-s + (1.99 + 3.46i)16-s + 5.73·17-s + (−2.36 + 2.36i)19-s + (2 + 0.535i)20-s + (−5.36 − 3.09i)22-s + (4.09 + 2.36i)23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.447 + 0.119i)5-s + (0.894 − 0.516i)7-s + (0.707 − 0.707i)8-s − 0.462i·10-s + (−0.341 + 1.27i)11-s + (0.250 + 0.933i)13-s + (0.267 + 0.997i)14-s + (0.499 + 0.866i)16-s + 1.39·17-s + (−0.542 + 0.542i)19-s + (0.447 + 0.119i)20-s + (−1.14 − 0.660i)22-s + (0.854 + 0.493i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639882 + 0.872690i\)
\(L(\frac12)\) \(\approx\) \(0.639882 + 0.872690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 0.267i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.36 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.13 - 4.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.901 - 3.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 + (2.36 - 2.36i)T - 19iT^{2} \)
23 \( 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.36 + 0.633i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.267 - 0.464i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.23 + 8.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.83 + 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.46 - 7.46i)T + 53iT^{2} \)
59 \( 1 + (7.33 - 1.96i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.1 - 3i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.76 + 6.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
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 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.34306667299370117469619772339, −10.30552204234646777914587189597, −9.568572917361450397718035081855, −8.450961128898838095579758063488, −7.54129978809340081514120468882, −7.15667263229433816755066088633, −5.76707694591779419618326178457, −4.72198208562042743276858694158, −3.88902546636047899246389647939, −1.55570902393976149217864041165, 0.861254593222668367361438668393, 2.60724110727375517500544306495, 3.65557597860058549489536714262, 4.95555497803790479961229531889, 5.82368261525715684172386052355, 7.76793829418102396565736011598, 8.201373802227012322701233265225, 9.030915615020235646337957301985, 10.18238836908735445679564023255, 11.12471554140359135741279122730

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