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Properties

Label 2-429-11.3-c1-0-14
Degree $2$
Conductor $429$
Sign $0.986 + 0.164i$
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  + (1.30 − 0.949i)2-s + (0.309 + 0.951i)3-s + (0.189 − 0.581i)4-s + (1.69 + 1.23i)5-s + (1.30 + 0.949i)6-s + (1.36 − 4.19i)7-s + (0.693 + 2.13i)8-s + (−0.809 + 0.587i)9-s + 3.39·10-s + (−3.28 − 0.467i)11-s + 0.611·12-s + (0.809 − 0.587i)13-s + (−2.20 − 6.77i)14-s + (−0.648 + 1.99i)15-s + (3.92 + 2.85i)16-s + (5.73 + 4.16i)17-s + ⋯
L(s)  = 1  + (0.924 − 0.671i)2-s + (0.178 + 0.549i)3-s + (0.0945 − 0.290i)4-s + (0.759 + 0.551i)5-s + (0.533 + 0.387i)6-s + (0.515 − 1.58i)7-s + (0.245 + 0.754i)8-s + (−0.269 + 0.195i)9-s + 1.07·10-s + (−0.990 − 0.140i)11-s + 0.176·12-s + (0.224 − 0.163i)13-s + (−0.588 − 1.81i)14-s + (−0.167 + 0.515i)15-s + (0.980 + 0.712i)16-s + (1.39 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.986 + 0.164i$
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: \((2,\ 429,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59478 - 0.214646i\)
\(L(\frac12)\) \(\approx\) \(2.59478 - 0.214646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.28 + 0.467i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-1.30 + 0.949i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.69 - 1.23i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.36 + 4.19i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-5.73 - 4.16i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.292 - 0.900i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + (-1.54 + 4.74i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.38 - 4.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.40 + 7.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.153 + 0.472i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (-0.374 - 1.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.886 - 0.644i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0285 - 0.0877i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.82 + 5.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.50T + 67T^{2} \)
71 \( 1 + (13.0 + 9.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.962 - 2.96i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.97 - 1.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.1 - 8.08i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.34T + 89T^{2} \)
97 \( 1 + (-6.01 + 4.37i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.84191865739064939593239255131, −10.52876888130070023060608817244, −9.898054056536804071137667147530, −8.177312660141891813080151979349, −7.64652832805363252663146688346, −6.05021826709917564867892992274, −5.12560454924233195559608018009, −4.01146734962103898300332614305, −3.30026566051567825040117742044, −1.89254318292998518000409331351, 1.70695732405322277530180459434, 3.05215542775130806175091641543, 4.93382404339380236846391693736, 5.46013428069670976976603649444, 6.08254177342015349860947416432, 7.38329212194877011630428175722, 8.283564786365659430077719207197, 9.303947791007853418278212661254, 10.06880442584469301007823217342, 11.64273808478987709218437843314

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