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Properties

Label 2-429-143.109-c1-0-2
Degree $2$
Conductor $429$
Sign $0.913 - 0.406i$
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  + (−1.06 − 1.06i)2-s − 3-s + 0.261i·4-s + (0.340 − 0.340i)5-s + (1.06 + 1.06i)6-s + (−0.593 + 0.593i)7-s + (−1.84 + 1.84i)8-s + 9-s − 0.724·10-s + (−1.96 + 2.67i)11-s − 0.261i·12-s + (2.47 + 2.61i)13-s + 1.26·14-s + (−0.340 + 0.340i)15-s + 4.45·16-s − 0.904·17-s + ⋯
L(s)  = 1  + (−0.751 − 0.751i)2-s − 0.577·3-s + 0.130i·4-s + (0.152 − 0.152i)5-s + (0.434 + 0.434i)6-s + (−0.224 + 0.224i)7-s + (−0.653 + 0.653i)8-s + 0.333·9-s − 0.229·10-s + (−0.591 + 0.806i)11-s − 0.0755i·12-s + (0.687 + 0.726i)13-s + 0.337·14-s + (−0.0880 + 0.0880i)15-s + 1.11·16-s − 0.219·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546470 + 0.116235i\)
\(L(\frac12)\) \(\approx\) \(0.546470 + 0.116235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + (1.96 - 2.67i)T \)
13 \( 1 + (-2.47 - 2.61i)T \)
good2 \( 1 + (1.06 + 1.06i)T + 2iT^{2} \)
5 \( 1 + (-0.340 + 0.340i)T - 5iT^{2} \)
7 \( 1 + (0.593 - 0.593i)T - 7iT^{2} \)
17 \( 1 + 0.904T + 17T^{2} \)
19 \( 1 + (2.48 + 2.48i)T + 19iT^{2} \)
23 \( 1 - 4.23iT - 23T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 + (-5.60 + 5.60i)T - 31iT^{2} \)
37 \( 1 + (-5.73 - 5.73i)T + 37iT^{2} \)
41 \( 1 + (-2.91 - 2.91i)T + 41iT^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (5.97 + 5.97i)T + 47iT^{2} \)
53 \( 1 - 3.19T + 53T^{2} \)
59 \( 1 + (5.71 + 5.71i)T + 59iT^{2} \)
61 \( 1 + 3.56iT - 61T^{2} \)
67 \( 1 + (10.0 - 10.0i)T - 67iT^{2} \)
71 \( 1 + (10.7 - 10.7i)T - 71iT^{2} \)
73 \( 1 + (10.3 - 10.3i)T - 73iT^{2} \)
79 \( 1 + 7.59iT - 79T^{2} \)
83 \( 1 + (-5.02 - 5.02i)T + 83iT^{2} \)
89 \( 1 + (-3.53 - 3.53i)T + 89iT^{2} \)
97 \( 1 + (-0.0569 + 0.0569i)T - 97iT^{2} \)
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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

−11.22993694250515631171796734049, −10.32339469522465694924247706384, −9.533848083037955328562282409155, −8.892289690552124142615273732181, −7.68216676830524119297368814167, −6.46046611097962125937683539840, −5.54417394941232179856281647506, −4.39249745528726929287614592449, −2.66695100747941639115622123839, −1.38789298552323150493952957639, 0.52515453497735591102369388273, 2.93486743470332520975707636217, 4.30736360814139222677783861333, 5.98385241816070497373139493360, 6.24470920022433893418237105987, 7.53340065249050296593281221924, 8.256360846596602486370494700326, 9.087887289075973408087446665782, 10.36064267334122991024174070988, 10.68306633286732821075525290210

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