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Properties

Label 2-429-429.311-c0-0-0
Degree $2$
Conductor $429$
Sign $-0.794 - 0.606i$
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  + (−1.30 + 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (0.5 + 0.363i)5-s + (−1.30 − 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (0.809 + 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (−0.190 + 0.587i)15-s + (0.499 − 1.53i)18-s + (0.809 − 0.587i)20-s − 1.61·22-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (0.5 + 0.363i)5-s + (−1.30 − 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (0.809 + 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (−0.190 + 0.587i)15-s + (0.499 − 1.53i)18-s + (0.809 − 0.587i)20-s − 1.61·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.794 - 0.606i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ -0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5178018221\)
\(L(\frac12)\) \(\approx\) \(0.5178018221\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−11.28762109256663416490852121132, −10.29600243377761459919900575565, −9.710361607604148672524999316250, −9.159561754712959830474060304949, −8.267743014963939036285783666670, −7.20908087491782886748791381401, −6.41294804193249137275589038306, −5.27858051588762843407143305543, −3.97555379983719071504035548865, −2.18858965324182768959442761440, 1.11081059766122049202867734004, 2.27925826571073484171743575601, 3.40355059272266370336791755239, 5.43197034432738427027019327933, 6.65782931712861500635309174840, 7.73551708271388336413552364642, 8.460915175321761235443975782793, 9.276596115350070564421554310108, 9.894486868715577257090248960921, 11.06943944990082376954400754315

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