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Properties

Label 2-429-11.4-c1-0-12
Degree $2$
Conductor $429$
Sign $-0.172 + 0.984i$
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  + (−1.44 − 1.04i)2-s + (−0.309 + 0.951i)3-s + (0.362 + 1.11i)4-s + (2.24 − 1.63i)5-s + (1.44 − 1.04i)6-s + (−0.0806 − 0.248i)7-s + (−0.454 + 1.39i)8-s + (−0.809 − 0.587i)9-s − 4.95·10-s + (0.650 − 3.25i)11-s − 1.17·12-s + (−0.809 − 0.587i)13-s + (−0.143 + 0.442i)14-s + (0.859 + 2.64i)15-s + (4.02 − 2.92i)16-s + (3.03 − 2.20i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.740i)2-s + (−0.178 + 0.549i)3-s + (0.181 + 0.558i)4-s + (1.00 − 0.731i)5-s + (0.588 − 0.427i)6-s + (−0.0304 − 0.0938i)7-s + (−0.160 + 0.494i)8-s + (−0.269 − 0.195i)9-s − 1.56·10-s + (0.196 − 0.980i)11-s − 0.339·12-s + (−0.224 − 0.163i)13-s + (−0.0384 + 0.118i)14-s + (0.221 + 0.682i)15-s + (1.00 − 0.730i)16-s + (0.736 − 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.172 + 0.984i$
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: \((2,\ 429,\ (\ :1/2),\ -0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534248 - 0.636110i\)
\(L(\frac12)\) \(\approx\) \(0.534248 - 0.636110i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.650 + 3.25i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.44 + 1.04i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.24 + 1.63i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.0806 + 0.248i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-3.03 + 2.20i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.50T + 23T^{2} \)
29 \( 1 + (2.17 + 6.69i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.33 + 5.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.58 + 7.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.24 + 3.81i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.53T + 43T^{2} \)
47 \( 1 + (-2.73 + 8.40i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.20 - 0.875i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.10 - 6.49i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.26 - 6.00i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.40T + 67T^{2} \)
71 \( 1 + (5.01 - 3.64i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.191 - 0.589i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.87 - 4.99i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.59 - 1.16i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-9.86 - 7.16i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.67108734278207395901662661188, −9.980772541286158109651020768710, −9.199344158621789585901164921117, −8.748976362915326623815845150023, −7.55200027250089991590566865483, −5.73669995026984098163183203496, −5.44087279579025795418397743350, −3.71708612220188466229653174015, −2.23805609432712055173317192982, −0.814556874584643315195165288122, 1.55824883534330690055867086184, 3.05189348141773058204820784909, 5.01863438517081674514019062094, 6.24551939885123933858300056415, 6.93715297819652880545325702979, 7.46192930974284438732562770640, 8.804044478878954122185468544104, 9.394479941729095349041993847940, 10.32272337179293563870986641748, 11.06852633319640967057447753733

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