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Properties

Label 2-429-429.389-c0-0-0
Degree $2$
Conductor $429$
Sign $-0.606 - 0.794i$
Analytic cond. $0.214098$
Root an. cond. $0.462708$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.690i)2-s + (−0.309 + 0.951i)3-s + (0.118 + 0.363i)4-s + (−1.53 + 1.11i)5-s + (−0.951 + 0.690i)6-s + (0.224 − 0.690i)8-s + (−0.809 − 0.587i)9-s − 2.23·10-s + (0.587 + 0.809i)11-s − 0.381·12-s + (0.809 + 0.587i)13-s + (−0.587 − 1.80i)15-s + (0.999 − 0.726i)16-s + (−0.363 − 1.11i)18-s + (−0.587 − 0.427i)20-s + ⋯
L(s)  = 1  + (0.951 + 0.690i)2-s + (−0.309 + 0.951i)3-s + (0.118 + 0.363i)4-s + (−1.53 + 1.11i)5-s + (−0.951 + 0.690i)6-s + (0.224 − 0.690i)8-s + (−0.809 − 0.587i)9-s − 2.23·10-s + (0.587 + 0.809i)11-s − 0.381·12-s + (0.809 + 0.587i)13-s + (−0.587 − 1.80i)15-s + (0.999 − 0.726i)16-s + (−0.363 − 1.11i)18-s + (−0.587 − 0.427i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.606 - 0.794i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ -0.606 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9831784913\)
\(L(\frac12)\) \(\approx\) \(0.9831784913\)
\(L(1)\) 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 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−11.70625812720386222765916687723, −10.91771712700696319199005670479, −10.12587100573526019792424533206, −8.982338493070922009312512390562, −7.69588932493201977110620130328, −6.80582416343585196375018629149, −6.11691347222716097898306595356, −4.67677027902836269375126871237, −4.05018168884220264028134805243, −3.30515647267295733169903165810, 1.20107535848543875899676196178, 3.13907700815883792100299171842, 4.04916246036323849635359222255, 5.08341753640254701040434817044, 6.07524348438055903842864879038, 7.55722538642061645412953070074, 8.242307319866917510318479018065, 8.873926369665993566651089229287, 10.97699382531054331634715745415, 11.31205680883836663969426250178

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