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Properties

Label 2-432-108.23-c1-0-15
Degree $2$
Conductor $432$
Sign $-0.668 + 0.743i$
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  + (0.991 + 1.42i)3-s + (−3.85 − 0.678i)5-s + (−1.90 − 2.26i)7-s + (−1.03 + 2.81i)9-s + (−0.560 − 3.17i)11-s + (1.43 − 0.523i)13-s + (−2.85 − 6.14i)15-s + (−2.60 − 1.50i)17-s + (−4.90 + 2.83i)19-s + (1.33 − 4.94i)21-s + (−6.11 − 5.13i)23-s + (9.66 + 3.51i)25-s + (−5.02 + 1.32i)27-s + (−1.57 + 4.33i)29-s + (6.11 − 7.28i)31-s + ⋯
L(s)  = 1  + (0.572 + 0.820i)3-s + (−1.72 − 0.303i)5-s + (−0.719 − 0.856i)7-s + (−0.344 + 0.938i)9-s + (−0.168 − 0.957i)11-s + (0.399 − 0.145i)13-s + (−0.736 − 1.58i)15-s + (−0.630 − 0.364i)17-s + (−1.12 + 0.649i)19-s + (0.291 − 1.08i)21-s + (−1.27 − 1.07i)23-s + (1.93 + 0.703i)25-s + (−0.967 + 0.254i)27-s + (−0.292 + 0.804i)29-s + (1.09 − 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124372 - 0.278920i\)
\(L(\frac12)\) \(\approx\) \(0.124372 - 0.278920i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.991 - 1.42i)T \)
good5 \( 1 + (3.85 + 0.678i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.90 + 2.26i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.560 + 3.17i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-1.43 + 0.523i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.60 + 1.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.90 - 2.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.11 + 5.13i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.57 - 4.33i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.11 + 7.28i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.24 - 3.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.210 - 0.577i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-3.35 + 0.591i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.982 + 0.824i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 5.01iT - 53T^{2} \)
59 \( 1 + (-0.0386 + 0.219i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (10.2 - 8.60i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.16 + 3.20i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.12 - 7.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.87 - 6.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.28 + 3.52i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.3 + 4.14i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.527 + 0.304i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.21 + 18.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.76888459479019345237505945592, −10.08449175925228559907262763578, −8.702832269889020514608316651997, −8.302287286080752584038629094261, −7.38211115545048614790168015474, −6.10380622677830062324807866800, −4.39338574560246170709999486707, −4.00402299819290920349423876281, −2.99174154274261465987935732728, −0.17198741092617762345982063227, 2.23253343378787386748513995867, 3.41867841449750803144150189162, 4.38355310375426971410376748689, 6.18655472620067462233185500692, 6.96247535099496511837902865931, 7.85119701958776656859948519335, 8.550560050174403973353486550004, 9.427662713511779375676780850080, 10.76355143670355861634557006362, 11.78965604828656040812878759130

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