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Properties

Label 2-429-13.12-c1-0-12
Degree $2$
Conductor $429$
Sign $0.686 - 0.726i$
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  + 0.409i·2-s + 3-s + 1.83·4-s + 4.13i·5-s + 0.409i·6-s − 5.18i·7-s + 1.56i·8-s + 9-s − 1.69·10-s − i·11-s + 1.83·12-s + (2.62 + 2.47i)13-s + 2.12·14-s + 4.13i·15-s + 3.02·16-s + 0.488·17-s + ⋯
L(s)  = 1  + 0.289i·2-s + 0.577·3-s + 0.916·4-s + 1.85i·5-s + 0.167i·6-s − 1.95i·7-s + 0.554i·8-s + 0.333·9-s − 0.535·10-s − 0.301i·11-s + 0.529·12-s + (0.726 + 0.686i)13-s + 0.566·14-s + 1.06i·15-s + 0.755·16-s + 0.118·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.686 - 0.726i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.686 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89098 + 0.814795i\)
\(L(\frac12)\) \(\approx\) \(1.89098 + 0.814795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + iT \)
13 \( 1 + (-2.62 - 2.47i)T \)
good2 \( 1 - 0.409iT - 2T^{2} \)
5 \( 1 - 4.13iT - 5T^{2} \)
7 \( 1 + 5.18iT - 7T^{2} \)
17 \( 1 - 0.488T + 17T^{2} \)
19 \( 1 + 0.446iT - 19T^{2} \)
23 \( 1 + 5.50T + 23T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 - 1.52iT - 31T^{2} \)
37 \( 1 - 7.87iT - 37T^{2} \)
41 \( 1 + 8.78iT - 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 + 5.42iT - 47T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 + 5.60iT - 59T^{2} \)
61 \( 1 + 0.855T + 61T^{2} \)
67 \( 1 + 3.87iT - 67T^{2} \)
71 \( 1 + 8.49iT - 71T^{2} \)
73 \( 1 + 5.95iT - 73T^{2} \)
79 \( 1 + 8.91T + 79T^{2} \)
83 \( 1 - 0.457iT - 83T^{2} \)
89 \( 1 - 4.68iT - 89T^{2} \)
97 \( 1 + 2.06iT - 97T^{2} \)
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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

−10.94539917339660671717066062068, −10.59758681941836505620175795665, −9.809039265442385578443775006852, −8.126334695173741803878068905162, −7.30699843148005084406340270271, −6.90039188600589836645595045539, −6.07092435989651048874288588739, −3.93032150257955636636312309353, −3.32304678488124777283055367981, −1.92720406763282712299682050200, 1.55963550244804990555737911146, 2.54996222963310855081426280688, 4.02474807516033968563846536951, 5.51567009356050232204051306507, 5.93039031644922263067601485338, 7.72971135938163635365889084529, 8.432035018970749257770489200325, 9.155458028874226036929297686485, 9.877838684132456377930739753728, 11.34625712363442729504635498876

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