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Properties

Label 2-432-108.59-c1-0-5
Degree $2$
Conductor $432$
Sign $0.881 - 0.471i$
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  + (1.35 − 1.07i)3-s + (1.16 + 3.20i)5-s + (−4.32 + 0.763i)7-s + (0.697 − 2.91i)9-s + (3.88 + 1.41i)11-s + (4.63 + 3.89i)13-s + (5.01 + 3.10i)15-s + (0.945 + 0.546i)17-s + (2.45 − 1.41i)19-s + (−5.06 + 5.68i)21-s + (0.327 − 1.85i)23-s + (−5.05 + 4.24i)25-s + (−2.18 − 4.71i)27-s + (−3.45 − 4.12i)29-s + (−1.52 − 0.268i)31-s + ⋯
L(s)  = 1  + (0.785 − 0.619i)3-s + (0.520 + 1.43i)5-s + (−1.63 + 0.288i)7-s + (0.232 − 0.972i)9-s + (1.17 + 0.426i)11-s + (1.28 + 1.07i)13-s + (1.29 + 0.800i)15-s + (0.229 + 0.132i)17-s + (0.562 − 0.324i)19-s + (−1.10 + 1.23i)21-s + (0.0681 − 0.386i)23-s + (−1.01 + 0.848i)25-s + (−0.419 − 0.907i)27-s + (−0.642 − 0.765i)29-s + (−0.273 − 0.0481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74618 + 0.437988i\)
\(L(\frac12)\) \(\approx\) \(1.74618 + 0.437988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.35 + 1.07i)T \)
good5 \( 1 + (-1.16 - 3.20i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (4.32 - 0.763i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-3.88 - 1.41i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-4.63 - 3.89i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.945 - 0.546i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.45 + 1.41i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.327 + 1.85i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.45 + 4.12i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.52 + 0.268i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.48 - 4.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 - 3.92i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.43 - 3.93i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.51 + 8.60i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + (3.09 - 1.12i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.463 + 2.62i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.32 - 2.77i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.842 + 1.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.58 + 9.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.26 - 7.46i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.240 + 0.201i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.60 + 3.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.903 - 0.328i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.33033859531463731266781237076, −9.984018695682134171780814388747, −9.511900967336719976166830442261, −8.674988201719209631931722771524, −7.17606212350023795481263127081, −6.49312853327414911628863276780, −6.24534544343088569550691426081, −3.75843596919828143242266295487, −3.15021167639863114685867594775, −1.87791413894661968333443236729, 1.23064618958256725927032933829, 3.28044528031705971151749696817, 3.88425912351632980744488802632, 5.36263507329391339255048262439, 6.15475790794760648088458998828, 7.57605528203042318173974241072, 8.899267329972668257786397341890, 9.047594366642495423322757223851, 9.926404668277797432621255650937, 10.79905049324743063618490275520

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