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Properties

Label 2-432-48.35-c1-0-26
Degree $2$
Conductor $432$
Sign $0.454 + 0.890i$
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  + (1.38 − 0.303i)2-s + (1.81 − 0.837i)4-s + (−1.39 − 1.39i)5-s − 1.33·7-s + (2.25 − 1.70i)8-s + (−2.34 − 1.50i)10-s + (4.37 − 4.37i)11-s + (1.20 + 1.20i)13-s + (−1.84 + 0.405i)14-s + (2.59 − 3.04i)16-s − 1.84i·17-s + (−1.19 + 1.19i)19-s + (−3.69 − 1.36i)20-s + (4.71 − 7.37i)22-s + 2.84i·23-s + ⋯
L(s)  = 1  + (0.976 − 0.214i)2-s + (0.907 − 0.418i)4-s + (−0.622 − 0.622i)5-s − 0.504·7-s + (0.796 − 0.603i)8-s + (−0.741 − 0.474i)10-s + (1.31 − 1.31i)11-s + (0.333 + 0.333i)13-s + (−0.493 + 0.108i)14-s + (0.648 − 0.760i)16-s − 0.448i·17-s + (−0.274 + 0.274i)19-s + (−0.825 − 0.304i)20-s + (1.00 − 1.57i)22-s + 0.592i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96073 - 1.20055i\)
\(L(\frac12)\) \(\approx\) \(1.96073 - 1.20055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.303i)T \)
3 \( 1 \)
good5 \( 1 + (1.39 + 1.39i)T + 5iT^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + (-4.37 + 4.37i)T - 11iT^{2} \)
13 \( 1 + (-1.20 - 1.20i)T + 13iT^{2} \)
17 \( 1 + 1.84iT - 17T^{2} \)
19 \( 1 + (1.19 - 1.19i)T - 19iT^{2} \)
23 \( 1 - 2.84iT - 23T^{2} \)
29 \( 1 + (0.485 - 0.485i)T - 29iT^{2} \)
31 \( 1 - 9.61iT - 31T^{2} \)
37 \( 1 + (6.99 - 6.99i)T - 37iT^{2} \)
41 \( 1 + 9.58T + 41T^{2} \)
43 \( 1 + (-5.79 - 5.79i)T + 43iT^{2} \)
47 \( 1 - 8.06T + 47T^{2} \)
53 \( 1 + (-3.25 - 3.25i)T + 53iT^{2} \)
59 \( 1 + (-0.730 + 0.730i)T - 59iT^{2} \)
61 \( 1 + (-7.32 - 7.32i)T + 61iT^{2} \)
67 \( 1 + (-7.38 + 7.38i)T - 67iT^{2} \)
71 \( 1 + 4.40iT - 71T^{2} \)
73 \( 1 + 0.385iT - 73T^{2} \)
79 \( 1 + 8.84iT - 79T^{2} \)
83 \( 1 + (-6.75 - 6.75i)T + 83iT^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 2.81T + 97T^{2} \)
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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

−11.35211910740532370830313569362, −10.33849033738252332277202449546, −9.124642897595295108329466575500, −8.324424082249125845632598322772, −6.93035977801590620723982568704, −6.22283890001532794193776289522, −5.09021945460704602855427505777, −3.94238326387887829361863529902, −3.23068500825380550035049424021, −1.24906127803376419866360823832, 2.16201983530846031006556346588, 3.63814805601233310772990456596, 4.20137396781872870920823844328, 5.63586716462471862986058292317, 6.78329122946631559910426173297, 7.14238791320893604058044565222, 8.368153638936933547162785121077, 9.628480418907425447177369000124, 10.67349030290939969223793980315, 11.49450289953460229881536250982

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