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Properties

Label 2-429-11.3-c1-0-1
Degree $2$
Conductor $429$
Sign $-0.964 + 0.265i$
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  + (−0.689 + 0.501i)2-s + (0.309 + 0.951i)3-s + (−0.393 + 1.21i)4-s + (−0.139 − 0.101i)5-s + (−0.689 − 0.501i)6-s + (−0.572 + 1.76i)7-s + (−0.862 − 2.65i)8-s + (−0.809 + 0.587i)9-s + 0.147·10-s + (−3.17 + 0.954i)11-s − 1.27·12-s + (−0.809 + 0.587i)13-s + (−0.488 − 1.50i)14-s + (0.0533 − 0.164i)15-s + (−0.134 − 0.0976i)16-s + (0.443 + 0.322i)17-s + ⋯
L(s)  = 1  + (−0.487 + 0.354i)2-s + (0.178 + 0.549i)3-s + (−0.196 + 0.605i)4-s + (−0.0625 − 0.0454i)5-s + (−0.281 − 0.204i)6-s + (−0.216 + 0.665i)7-s + (−0.304 − 0.938i)8-s + (−0.269 + 0.195i)9-s + 0.0465·10-s + (−0.957 + 0.287i)11-s − 0.367·12-s + (−0.224 + 0.163i)13-s + (−0.130 − 0.401i)14-s + (0.0137 − 0.0424i)15-s + (−0.0336 − 0.0244i)16-s + (0.107 + 0.0781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0732130 - 0.542505i\)
\(L(\frac12)\) \(\approx\) \(0.0732130 - 0.542505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.17 - 0.954i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.689 - 0.501i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.139 + 0.101i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.572 - 1.76i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-0.443 - 0.322i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.849 - 2.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.38T + 23T^{2} \)
29 \( 1 + (-1.91 + 5.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.83 - 4.96i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.34 - 10.3i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.05 + 3.24i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + (2.04 + 6.30i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.87 - 2.09i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.02 - 12.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.04 - 1.48i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.221T + 67T^{2} \)
71 \( 1 + (-5.52 - 4.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.19 - 9.84i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.20 + 1.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.18 - 5.94i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 + (-12.5 + 9.12i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.81475374955300927478846866857, −10.37185965311669056697156697033, −9.806094297760616931817874075697, −8.799602108170843087888274229383, −8.172619991696436379907288222582, −7.28152116550127100704296090094, −6.04311511180178941923383278311, −4.87787441520769056114452709407, −3.70769063574871932498160743359, −2.53801852473773518691172048558, 0.38315610418072010641204690504, 1.96254734537792728337250424398, 3.33202122206070753925433857582, 4.97517212125712486009733598444, 5.88499388448316631431560057544, 7.15806106747909832049289842314, 7.914442149081173216002905769943, 9.013753924073649219525756983956, 9.748230061285372935871114882641, 10.78671136310961837536836829002

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