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Properties

Label 2-429-11.3-c1-0-18
Degree $2$
Conductor $429$
Sign $-0.953 + 0.299i$
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  + (−2.05 + 1.49i)2-s + (−0.309 − 0.951i)3-s + (1.37 − 4.22i)4-s + (−3.43 − 2.49i)5-s + (2.05 + 1.49i)6-s + (0.899 − 2.76i)7-s + (1.91 + 5.89i)8-s + (−0.809 + 0.587i)9-s + 10.7·10-s + (3.13 − 1.06i)11-s − 4.44·12-s + (−0.809 + 0.587i)13-s + (2.28 + 7.02i)14-s + (−1.31 + 4.03i)15-s + (−5.54 − 4.02i)16-s + (−0.626 − 0.455i)17-s + ⋯
L(s)  = 1  + (−1.45 + 1.05i)2-s + (−0.178 − 0.549i)3-s + (0.686 − 2.11i)4-s + (−1.53 − 1.11i)5-s + (0.838 + 0.609i)6-s + (0.339 − 1.04i)7-s + (0.677 + 2.08i)8-s + (−0.269 + 0.195i)9-s + 3.40·10-s + (0.946 − 0.322i)11-s − 1.28·12-s + (−0.224 + 0.163i)13-s + (0.610 + 1.87i)14-s + (−0.338 + 1.04i)15-s + (−1.38 − 1.00i)16-s + (−0.151 − 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.953 + 0.299i$
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: \((2,\ 429,\ (\ :1/2),\ -0.953 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0332030 - 0.216268i\)
\(L(\frac12)\) \(\approx\) \(0.0332030 - 0.216268i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.13 + 1.06i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (2.05 - 1.49i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.43 + 2.49i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.899 + 2.76i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (0.626 + 0.455i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.41 + 7.43i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 + (0.289 - 0.890i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.28 - 2.38i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.44 - 4.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.615 + 1.89i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.54T + 43T^{2} \)
47 \( 1 + (1.43 + 4.41i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.46 - 4.69i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.90 + 8.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.52 - 3.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + (-8.27 - 6.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.616 - 1.89i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.01 - 3.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.9 - 10.1i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.56T + 89T^{2} \)
97 \( 1 + (7.32 - 5.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.80277025481106529586833810982, −9.368327905185049451959224475206, −8.672880911504578964648411458045, −8.014626444974708546385619027140, −7.21550260831549397351124489940, −6.67066382395773080833420421963, −5.11368285726994737456734492775, −4.07398149839304852392607400235, −1.25262685501100019587267549419, −0.25923984184246025857751771984, 2.11687363152888771162886307939, 3.38913369470814025824140682155, 4.14043106424080389315064826956, 6.18849741892514713687414015511, 7.47657371456933253873065047738, 8.137758513481496113729054787596, 8.949358224473799592046727497418, 9.933837746455359967115427593704, 10.70876814149331464688261819884, 11.37129486839997507220594377699

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