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Properties

Label 2-432-48.35-c1-0-29
Degree $2$
Conductor $432$
Sign $-0.310 - 0.950i$
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  + (−0.762 − 1.19i)2-s + (−0.835 + 1.81i)4-s + (−3.06 − 3.06i)5-s + 1.75·7-s + (2.80 − 0.390i)8-s + (−1.31 + 5.98i)10-s + (−2.61 + 2.61i)11-s + (−3.12 − 3.12i)13-s + (−1.34 − 2.09i)14-s + (−2.60 − 3.03i)16-s + 0.448i·17-s + (−2.39 + 2.39i)19-s + (8.12 − 3.00i)20-s + (5.11 + 1.12i)22-s + 6.52i·23-s + ⋯
L(s)  = 1  + (−0.539 − 0.842i)2-s + (−0.417 + 0.908i)4-s + (−1.36 − 1.36i)5-s + 0.665·7-s + (0.990 − 0.138i)8-s + (−0.414 + 1.89i)10-s + (−0.789 + 0.789i)11-s + (−0.866 − 0.866i)13-s + (−0.358 − 0.559i)14-s + (−0.650 − 0.759i)16-s + 0.108i·17-s + (−0.549 + 0.549i)19-s + (1.81 − 0.671i)20-s + (1.09 + 0.238i)22-s + 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.310 - 0.950i$
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: \((2,\ 432,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0268171 + 0.0369729i\)
\(L(\frac12)\) \(\approx\) \(0.0268171 + 0.0369729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.762 + 1.19i)T \)
3 \( 1 \)
good5 \( 1 + (3.06 + 3.06i)T + 5iT^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 + (2.61 - 2.61i)T - 11iT^{2} \)
13 \( 1 + (3.12 + 3.12i)T + 13iT^{2} \)
17 \( 1 - 0.448iT - 17T^{2} \)
19 \( 1 + (2.39 - 2.39i)T - 19iT^{2} \)
23 \( 1 - 6.52iT - 23T^{2} \)
29 \( 1 + (-1.11 + 1.11i)T - 29iT^{2} \)
31 \( 1 + 5.03iT - 31T^{2} \)
37 \( 1 + (6.18 - 6.18i)T - 37iT^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 + (0.0280 + 0.0280i)T + 43iT^{2} \)
47 \( 1 + 0.0301T + 47T^{2} \)
53 \( 1 + (-1.25 - 1.25i)T + 53iT^{2} \)
59 \( 1 + (2.95 - 2.95i)T - 59iT^{2} \)
61 \( 1 + (1.22 + 1.22i)T + 61iT^{2} \)
67 \( 1 + (-4.25 + 4.25i)T - 67iT^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (4.30 + 4.30i)T + 83iT^{2} \)
89 \( 1 - 0.159T + 89T^{2} \)
97 \( 1 + 6.07T + 97T^{2} \)
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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

−10.54765302019164606176689328365, −9.658025244752677628067437506023, −8.625184945718735176231391512203, −7.82918824275865185921775434594, −7.56573801768759543909250480290, −5.13107389388201039620357182803, −4.58374428863529267680977510734, −3.41357335570984313702848255329, −1.69058720558483569839232442702, −0.03431640524961979217212043264, 2.57327912385558517864099717410, 4.11770151510938079664936778979, 5.13879379557707724422489313054, 6.69243746156768664261589242982, 7.07777725294837463470902330346, 8.124070956063151731987944443592, 8.625722783166029303005268836711, 10.13454428135155132568390414589, 10.85721967530659250075319746593, 11.37806534721500949847453553755

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