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Properties

Label 2-429-11.4-c1-0-10
Degree $2$
Conductor $429$
Sign $0.228 + 0.973i$
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  + (−2.26 − 1.64i)2-s + (−0.309 + 0.951i)3-s + (1.80 + 5.54i)4-s + (2.68 − 1.95i)5-s + (2.26 − 1.64i)6-s + (−0.449 − 1.38i)7-s + (3.31 − 10.1i)8-s + (−0.809 − 0.587i)9-s − 9.28·10-s + (0.682 + 3.24i)11-s − 5.83·12-s + (0.809 + 0.587i)13-s + (−1.25 + 3.86i)14-s + (1.02 + 3.15i)15-s + (−14.8 + 10.7i)16-s + (5.09 − 3.70i)17-s + ⋯
L(s)  = 1  + (−1.60 − 1.16i)2-s + (−0.178 + 0.549i)3-s + (0.901 + 2.77i)4-s + (1.20 − 0.872i)5-s + (0.924 − 0.671i)6-s + (−0.169 − 0.522i)7-s + (1.17 − 3.60i)8-s + (−0.269 − 0.195i)9-s − 2.93·10-s + (0.205 + 0.978i)11-s − 1.68·12-s + (0.224 + 0.163i)13-s + (−0.335 + 1.03i)14-s + (0.264 + 0.815i)15-s + (−3.71 + 2.69i)16-s + (1.23 − 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.228 + 0.973i$
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: \((2,\ 429,\ (\ :1/2),\ 0.228 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592313 - 0.469177i\)
\(L(\frac12)\) \(\approx\) \(0.592313 - 0.469177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.682 - 3.24i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (2.26 + 1.64i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.68 + 1.95i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.449 + 1.38i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-5.09 + 3.70i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.174 + 0.537i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 + (0.493 + 1.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.96 - 5.78i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.422 + 1.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.12 + 3.47i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.12T + 43T^{2} \)
47 \( 1 + (-1.75 + 5.41i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.89 - 1.37i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.82 + 8.68i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.21 + 1.60i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.27T + 67T^{2} \)
71 \( 1 + (0.143 - 0.104i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.01 + 3.11i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.0831 - 0.0603i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.2 + 8.19i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.06T + 89T^{2} \)
97 \( 1 + (-11.8 - 8.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.46481482652928011975663529281, −10.00163085926394836541197584927, −9.482723985119717842913166604697, −8.728840565744443136945654529288, −7.66546343903675210909890276588, −6.56627563632243746672126856632, −4.92495746773500330694673619732, −3.65020428882172433960295392113, −2.21690021937030744643466605786, −0.996872940391275884506708284584, 1.29843515128207536141016673227, 2.59153326446434694038300931785, 5.63671439543295186353824659678, 6.00114942313065243293797232256, 6.61436365141599374968178595361, 7.80190797424140161147225893287, 8.458226320623120945178705196328, 9.512749605008167418791407668491, 10.17504512775640879111993946707, 10.86349060905718538421963950323

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