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Properties

Label 2-432-108.11-c1-0-9
Degree $2$
Conductor $432$
Sign $0.925 - 0.378i$
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  + (1.64 − 0.550i)3-s + (−1.00 + 2.76i)5-s + (0.201 + 0.0355i)7-s + (2.39 − 1.80i)9-s + (2.91 − 1.05i)11-s + (1.23 − 1.03i)13-s + (−0.130 + 5.09i)15-s + (−2.86 + 1.65i)17-s + (5.98 + 3.45i)19-s + (0.350 − 0.0525i)21-s + (1.21 + 6.89i)23-s + (−2.79 − 2.34i)25-s + (2.93 − 4.28i)27-s + (−2.70 + 3.22i)29-s + (−0.173 + 0.0305i)31-s + ⋯
L(s)  = 1  + (0.948 − 0.317i)3-s + (−0.449 + 1.23i)5-s + (0.0760 + 0.0134i)7-s + (0.797 − 0.602i)9-s + (0.877 − 0.319i)11-s + (0.342 − 0.287i)13-s + (−0.0336 + 1.31i)15-s + (−0.694 + 0.401i)17-s + (1.37 + 0.792i)19-s + (0.0764 − 0.0114i)21-s + (0.253 + 1.43i)23-s + (−0.559 − 0.469i)25-s + (0.564 − 0.825i)27-s + (−0.503 + 0.599i)29-s + (−0.0311 + 0.00548i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.925 - 0.378i$
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: \((2,\ 432,\ (\ :1/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83877 + 0.361946i\)
\(L(\frac12)\) \(\approx\) \(1.83877 + 0.361946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.550i)T \)
good5 \( 1 + (1.00 - 2.76i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.201 - 0.0355i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.91 + 1.05i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.23 + 1.03i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.86 - 1.65i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.98 - 3.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.21 - 6.89i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.70 - 3.22i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.173 - 0.0305i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (5.46 + 9.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.982 + 1.17i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.57 + 9.83i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.76 + 9.98i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.56iT - 53T^{2} \)
59 \( 1 + (4.69 + 1.71i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.18 + 6.69i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.60 + 3.10i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.61 - 4.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.68 - 8.11i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.13 - 7.31i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.47 + 7.10i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-2.64 - 1.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.58 - 0.577i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.24249059332575219962976368912, −10.29511712554824505010542126642, −9.309465189569863463320053237328, −8.452194158735912212109503610158, −7.35792557277206164492445004627, −6.93274385766506706465617985090, −5.65145227100512272930978429995, −3.69860478246026821800670151872, −3.37146274638413071891098778087, −1.76014425560101709044188695889, 1.35635775811634950409251378580, 3.02765889873010073125619366014, 4.40470713699698319555594516031, 4.80703583757921108857200851901, 6.56200420865343843756964615087, 7.64447349105714577775366819500, 8.585676176916799812940239188516, 9.123862253187246346264442789924, 9.857953267700444997560678574234, 11.21315375665235255169585589789

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