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Properties

Label 2-429-11.3-c1-0-13
Degree $2$
Conductor $429$
Sign $0.873 + 0.487i$
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  + (−2.09 + 1.52i)2-s + (0.309 + 0.951i)3-s + (1.44 − 4.45i)4-s + (0.296 + 0.215i)5-s + (−2.09 − 1.52i)6-s + (0.607 − 1.86i)7-s + (2.14 + 6.60i)8-s + (−0.809 + 0.587i)9-s − 0.949·10-s + (−2.93 − 1.54i)11-s + 4.68·12-s + (0.809 − 0.587i)13-s + (1.56 + 4.83i)14-s + (−0.113 + 0.349i)15-s + (−6.95 − 5.05i)16-s + (−5.32 − 3.86i)17-s + ⋯
L(s)  = 1  + (−1.47 + 1.07i)2-s + (0.178 + 0.549i)3-s + (0.724 − 2.22i)4-s + (0.132 + 0.0964i)5-s + (−0.854 − 0.620i)6-s + (0.229 − 0.706i)7-s + (0.759 + 2.33i)8-s + (−0.269 + 0.195i)9-s − 0.300·10-s + (−0.884 − 0.466i)11-s + 1.35·12-s + (0.224 − 0.163i)13-s + (0.419 + 1.29i)14-s + (−0.0292 + 0.0901i)15-s + (−1.73 − 1.26i)16-s + (−1.29 − 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.873 + 0.487i$
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: \((2,\ 429,\ (\ :1/2),\ 0.873 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448567 - 0.116780i\)
\(L(\frac12)\) \(\approx\) \(0.448567 - 0.116780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.93 + 1.54i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (2.09 - 1.52i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.296 - 0.215i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.607 + 1.86i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (5.32 + 3.86i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.43 + 7.50i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 + (-0.628 + 1.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.68 - 5.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0969 - 0.298i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.23 + 3.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 + (-1.28 - 3.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.75 + 6.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.576 - 1.77i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.17 - 5.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.34T + 67T^{2} \)
71 \( 1 + (3.80 + 2.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.692 - 2.13i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.09 - 2.24i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.7 + 7.78i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.86T + 89T^{2} \)
97 \( 1 + (-9.81 + 7.13i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.85792654109862250648241646725, −10.03973859047549708692052959030, −8.920880644618681347748725086075, −8.653363102906118376903415684048, −7.35937535253234771245718221281, −6.86229441798485548473120844120, −5.57228069331868970418042368405, −4.58238684499932474893302509558, −2.54784520136824288819053619137, −0.45313237011715163543378916506, 1.67777235415862953596833948094, 2.40701060820222871182270853809, 3.83423262904107117743389122226, 5.67514282533270180222322732970, 7.05105102574010782406374905081, 7.961912664182403877507901616360, 8.665435496124540752867130056177, 9.325213622023473891304052482404, 10.40684953350319696638657130557, 11.04576790627859888382749290948

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