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Properties

Label 2-429-11.5-c1-0-6
Degree $2$
Conductor $429$
Sign $-0.762 - 0.646i$
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  + (0.400 + 1.23i)2-s + (0.809 + 0.587i)3-s + (0.259 − 0.188i)4-s + (−0.412 + 1.27i)5-s + (−0.400 + 1.23i)6-s + (−4.15 + 3.01i)7-s + (2.43 + 1.76i)8-s + (0.309 + 0.951i)9-s − 1.73·10-s + (−3.12 − 1.11i)11-s + 0.320·12-s + (0.309 + 0.951i)13-s + (−5.38 − 3.91i)14-s + (−1.08 + 0.785i)15-s + (−1.00 + 3.09i)16-s + (1.60 − 4.94i)17-s + ⋯
L(s)  = 1  + (0.283 + 0.871i)2-s + (0.467 + 0.339i)3-s + (0.129 − 0.0943i)4-s + (−0.184 + 0.568i)5-s + (−0.163 + 0.503i)6-s + (−1.56 + 1.14i)7-s + (0.860 + 0.624i)8-s + (0.103 + 0.317i)9-s − 0.547·10-s + (−0.941 − 0.336i)11-s + 0.0926·12-s + (0.0857 + 0.263i)13-s + (−1.43 − 1.04i)14-s + (−0.278 + 0.202i)15-s + (−0.251 + 0.773i)16-s + (0.389 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.762 - 0.646i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572606 + 1.56132i\)
\(L(\frac12)\) \(\approx\) \(0.572606 + 1.56132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.12 + 1.11i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.400 - 1.23i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.412 - 1.27i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (4.15 - 3.01i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-1.60 + 4.94i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.42 - 2.49i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + (3.80 - 2.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.31 - 4.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.84 + 4.97i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.36 + 4.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.51T + 43T^{2} \)
47 \( 1 + (-9.33 - 6.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.40 + 10.4i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.836 + 0.607i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.169 - 0.522i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.62T + 67T^{2} \)
71 \( 1 + (-1.34 + 4.13i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.50 - 6.17i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.67 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.30 + 7.10i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 7.64T + 89T^{2} \)
97 \( 1 + (1.81 + 5.59i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.43696611731905474572038107269, −10.50488324411084788659537147911, −9.593599679319782897794871798507, −8.802753598484534027621195907810, −7.54843699289567781900862080722, −6.89529476885648314611888705882, −5.81173346201776343424501612251, −5.16053953017397121195970166082, −3.32956949294661048184378488178, −2.60689099432501869229518210990, 0.938092608058058990400108777904, 2.70796348667968050000681676597, 3.52631564085026795909249441888, 4.51572734677466214307422848954, 6.20044617886659245742478048348, 7.27519868570247615255735050129, 7.86350619008972710377835661474, 9.254866401034883798516658847593, 10.15621246220477908400541296249, 10.66218042153020569596616913139

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