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Properties

Label 2-429-11.4-c1-0-0
Degree $2$
Conductor $429$
Sign $-0.588 - 0.808i$
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  + (−0.214 − 0.156i)2-s + (0.309 − 0.951i)3-s + (−0.596 − 1.83i)4-s + (−3.10 + 2.25i)5-s + (−0.214 + 0.156i)6-s + (−0.868 − 2.67i)7-s + (−0.322 + 0.992i)8-s + (−0.809 − 0.587i)9-s + 1.02·10-s + (1.88 + 2.72i)11-s − 1.92·12-s + (0.809 + 0.587i)13-s + (−0.230 + 0.709i)14-s + (1.18 + 3.65i)15-s + (−2.89 + 2.10i)16-s + (−2.81 + 2.04i)17-s + ⋯
L(s)  = 1  + (−0.151 − 0.110i)2-s + (0.178 − 0.549i)3-s + (−0.298 − 0.917i)4-s + (−1.39 + 1.01i)5-s + (−0.0877 + 0.0637i)6-s + (−0.328 − 1.01i)7-s + (−0.114 + 0.350i)8-s + (−0.269 − 0.195i)9-s + 0.322·10-s + (0.568 + 0.822i)11-s − 0.556·12-s + (0.224 + 0.163i)13-s + (−0.0616 + 0.189i)14-s + (0.306 + 0.943i)15-s + (−0.724 + 0.526i)16-s + (−0.683 + 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0297937 + 0.0584969i\)
\(L(\frac12)\) \(\approx\) \(0.0297937 + 0.0584969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 2.72i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.214 + 0.156i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (3.10 - 2.25i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.868 + 2.67i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (2.81 - 2.04i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.45 - 4.46i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.27T + 23T^{2} \)
29 \( 1 + (-0.784 - 2.41i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.85 + 2.80i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.99 - 9.21i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.26 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.76T + 43T^{2} \)
47 \( 1 + (-1.73 + 5.32i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.20 + 5.96i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.599 - 1.84i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.435 - 0.316i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.204T + 67T^{2} \)
71 \( 1 + (7.34 - 5.33i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.24 + 13.0i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.2 + 7.45i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (14.3 - 10.4i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-11.3 - 8.24i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.43510809688004351483419126055, −10.47878115621962913460720505456, −9.998375476428226311501355435728, −8.623758372044833113060166191505, −7.70342044769549105829466373870, −6.86283005622930703560067614300, −6.18602246643933087024406927596, −4.31749232137689014381068377998, −3.68319754315846823158876071599, −1.84389601754726470907132064098, 0.04228037465564830460371285573, 2.91349398634755514518844833141, 3.97706333201726809780010599614, 4.65505121993265933179576371254, 6.05050228496216823159329364670, 7.49261905401711899657983830573, 8.396304264083528417222811622256, 8.831218298447388055948704298577, 9.478748308874694516098649211659, 11.25146922506259775534218500496

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