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Properties

Label 2-429-11.4-c1-0-14
Degree $2$
Conductor $429$
Sign $0.858 - 0.513i$
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  + (1.72 + 1.25i)2-s + (0.309 − 0.951i)3-s + (0.784 + 2.41i)4-s + (1.18 − 0.863i)5-s + (1.72 − 1.25i)6-s + (0.349 + 1.07i)7-s + (−0.354 + 1.09i)8-s + (−0.809 − 0.587i)9-s + 3.13·10-s + (2.88 − 1.63i)11-s + 2.53·12-s + (−0.809 − 0.587i)13-s + (−0.744 + 2.29i)14-s + (−0.454 − 1.39i)15-s + (2.12 − 1.54i)16-s + (−5.36 + 3.89i)17-s + ⋯
L(s)  = 1  + (1.21 + 0.885i)2-s + (0.178 − 0.549i)3-s + (0.392 + 1.20i)4-s + (0.531 − 0.386i)5-s + (0.703 − 0.511i)6-s + (0.132 + 0.406i)7-s + (−0.125 + 0.386i)8-s + (−0.269 − 0.195i)9-s + 0.989·10-s + (0.870 − 0.492i)11-s + 0.732·12-s + (−0.224 − 0.163i)13-s + (−0.198 + 0.612i)14-s + (−0.117 − 0.360i)15-s + (0.532 − 0.386i)16-s + (−1.30 + 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.858 - 0.513i$
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: \((2,\ 429,\ (\ :1/2),\ 0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.83395 + 0.782612i\)
\(L(\frac12)\) \(\approx\) \(2.83395 + 0.782612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.88 + 1.63i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.72 - 1.25i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.18 + 0.863i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.349 - 1.07i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (5.36 - 3.89i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.893 - 2.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.09T + 23T^{2} \)
29 \( 1 + (0.199 + 0.613i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.03 - 1.48i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.44 + 4.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.218 + 0.673i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + (-0.553 + 1.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.217 - 0.158i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.821 - 2.52i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.42 - 4.66i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + (-7.04 + 5.12i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.19 + 6.74i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (13.0 + 9.46i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.69 - 2.68i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 + (-8.14 - 5.91i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.73547072722416817633207770292, −10.38959273728640187042704726512, −9.098639636853467569183234647024, −8.356552914350047617236629537959, −7.21992067832715146477458000135, −6.24553856674979825661867920957, −5.77097007459125030609436658335, −4.57667498904112307465067748260, −3.49362557292877988112989781096, −1.83043756770402847031616112271, 1.98460841758114389702726430102, 3.00630016384149098916706672806, 4.29001388454053726226003139611, 4.76191450255286375867900100646, 6.12153448185297610967461661736, 7.05706191236253548578255757179, 8.599043371159894436495784446831, 9.671240609847016968156131782159, 10.35635756217461178646867480577, 11.31606026481509739778622207316

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