Properties

Label 2-429-11.4-c1-0-2
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: $\chi_{429} (235, \cdot )$
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

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

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