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Properties

Label 2-429-429.38-c0-0-3
Degree $2$
Conductor $429$
Sign $-0.624 + 0.781i$
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  + (−0.190 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.5 − 0.363i)4-s + (0.5 − 1.53i)5-s + (−0.190 + 0.587i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s + (−0.309 + 0.951i)11-s − 0.618·12-s + (0.309 + 0.951i)13-s + (−1.30 + 0.951i)15-s + (0.5 − 0.363i)18-s + (−0.309 − 0.951i)20-s + 0.618·22-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.5 − 0.363i)4-s + (0.5 − 1.53i)5-s + (−0.190 + 0.587i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s + (−0.309 + 0.951i)11-s − 0.618·12-s + (0.309 + 0.951i)13-s + (−1.30 + 0.951i)15-s + (0.5 − 0.363i)18-s + (−0.309 − 0.951i)20-s + 0.618·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ -0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7193698799\)
\(L(\frac12)\) \(\approx\) \(0.7193698799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−11.29438079843982295421005890941, −10.13289418182478511531048708088, −9.523297056904322738493104937906, −8.471958572386302409256479246537, −7.20877837705509464941076740476, −6.25518012263535103592175532361, −5.34198176815464359448850852181, −4.42425776941067545086850175368, −2.16217725319526669289692058913, −1.27280001683882914027761634588, 2.73249382940162996800094119073, 3.59862104190831343971752432541, 5.54001912774411378084669632374, 6.07804249518703855234572348331, 6.88690813531367525138184837542, 7.80364615787510018936144533052, 8.990393077586004577456300244829, 10.23520286515790237639414073594, 10.79421015165962637378228418906, 11.35197267811306268530763204642

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