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Properties

Label 2-429-11.3-c1-0-23
Degree $2$
Conductor $429$
Sign $-0.904 + 0.427i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
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  + (2.10 − 1.52i)2-s + (−0.309 − 0.951i)3-s + (1.47 − 4.53i)4-s + (−1.86 − 1.35i)5-s + (−2.10 − 1.52i)6-s + (−0.646 + 1.99i)7-s + (−2.22 − 6.84i)8-s + (−0.809 + 0.587i)9-s − 5.99·10-s + (2.96 − 1.48i)11-s − 4.76·12-s + (−0.809 + 0.587i)13-s + (1.68 + 5.17i)14-s + (−0.712 + 2.19i)15-s + (−7.42 − 5.39i)16-s + (4.25 + 3.08i)17-s + ⋯
L(s)  = 1  + (1.48 − 1.08i)2-s + (−0.178 − 0.549i)3-s + (0.736 − 2.26i)4-s + (−0.834 − 0.606i)5-s + (−0.859 − 0.624i)6-s + (−0.244 + 0.752i)7-s + (−0.786 − 2.41i)8-s + (−0.269 + 0.195i)9-s − 1.89·10-s + (0.893 − 0.448i)11-s − 1.37·12-s + (−0.224 + 0.163i)13-s + (0.449 + 1.38i)14-s + (−0.184 + 0.566i)15-s + (−1.85 − 1.34i)16-s + (1.03 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.904 + 0.427i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546790 - 2.43546i\)
\(L(\frac12)\) \(\approx\) \(0.546790 - 2.43546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.96 + 1.48i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-2.10 + 1.52i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.86 + 1.35i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.646 - 1.99i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-4.25 - 3.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.875 + 2.69i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.893T + 23T^{2} \)
29 \( 1 + (-1.08 + 3.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.81 + 2.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.10 - 3.41i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.761 - 2.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 + (-2.77 - 8.54i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.06 + 6.58i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.50 - 13.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.72 + 7.06i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (5.69 + 4.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.47 - 7.61i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.29 - 0.938i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.29 + 3.12i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (13.8 - 10.0i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37129652431127463827019854443, −10.25206178778773382114378150399, −9.111382089260483308874060980119, −8.019996471459824186599240011303, −6.52351598388337015504146251897, −5.79619805194585127650253417671, −4.71136914447481850344647813637, −3.77912238509531046116375049909, −2.63622323360698819264049029720, −1.14593063665606775519267360783, 3.21704527772667274083850180918, 3.82196357726360097356795396913, 4.72392235880433720314611074117, 5.77640276899881534972836834272, 6.98756976481197448602696938999, 7.26527131264344636168537822250, 8.418916779084012544483734681910, 9.857414932841800950665476176279, 10.90111556946693473149756385433, 11.99652178428764355701491978003

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