Properties

Label 2-429-11.4-c1-0-23
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: $\chi_{429} (235, \cdot )$
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

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

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