Properties

Label 2-432-108.11-c1-0-2
Degree $2$
Conductor $432$
Sign $-0.327 - 0.944i$
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  + (0.0760 − 1.73i)3-s + (−1.38 + 3.79i)5-s + (1.08 + 0.192i)7-s + (−2.98 − 0.263i)9-s + (−5.69 + 2.07i)11-s + (−2.30 + 1.93i)13-s + (6.46 + 2.68i)15-s + (−2.73 + 1.57i)17-s + (3.28 + 1.89i)19-s + (0.415 − 1.87i)21-s + (−0.347 − 1.96i)23-s + (−8.69 − 7.29i)25-s + (−0.683 + 5.15i)27-s + (2.14 − 2.56i)29-s + (3.05 − 0.538i)31-s + ⋯
L(s)  = 1  + (0.0439 − 0.999i)3-s + (−0.618 + 1.69i)5-s + (0.411 + 0.0725i)7-s + (−0.996 − 0.0877i)9-s + (−1.71 + 0.624i)11-s + (−0.639 + 0.536i)13-s + (1.67 + 0.692i)15-s + (−0.663 + 0.382i)17-s + (0.753 + 0.435i)19-s + (0.0906 − 0.408i)21-s + (−0.0723 − 0.410i)23-s + (−1.73 − 1.45i)25-s + (−0.131 + 0.991i)27-s + (0.399 − 0.475i)29-s + (0.548 − 0.0967i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.327 - 0.944i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399855 + 0.561753i\)
\(L(\frac12)\) \(\approx\) \(0.399855 + 0.561753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0760 + 1.73i)T \)
good5 \( 1 + (1.38 - 3.79i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.08 - 0.192i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (5.69 - 2.07i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (2.30 - 1.93i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.28 - 1.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.347 + 1.96i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.14 + 2.56i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.05 + 0.538i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.96 - 8.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.58 - 6.66i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.77 + 4.86i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.132 - 0.753i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.19iT - 53T^{2} \)
59 \( 1 + (-7.48 - 2.72i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.13 - 6.45i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.17 + 3.78i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.30 + 7.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.12 + 7.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.29 - 5.11i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.12 + 7.65i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-10.9 - 6.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.13 - 2.23i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.49578747593066544957872588035, −10.65859562152714186141204348281, −9.879459483297531773835312111439, −8.191825304916158530174039127981, −7.68188553127055596243160570229, −6.95461118730327459635483449648, −6.06868934519575632955011859574, −4.64094999859575987513968027009, −2.96961876376100852768160943072, −2.27047201022877222927516033441, 0.41000137217625594894629246921, 2.78572209053516488803841899550, 4.21004143413001615392150212629, 5.11911490608867819856004707747, 5.45392967806679208385325879053, 7.63397211383011884759086724138, 8.235715084521292977408325200319, 9.032494689554122829530994043206, 9.881945257047727827331841572193, 10.95022407133950008935731088890

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