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Properties

Label 2-429-11.5-c1-0-5
Degree $2$
Conductor $429$
Sign $-0.793 + 0.608i$
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  + (0.856 + 2.63i)2-s + (0.809 + 0.587i)3-s + (−4.59 + 3.34i)4-s + (0.276 − 0.850i)5-s + (−0.856 + 2.63i)6-s + (−1.00 + 0.728i)7-s + (−8.25 − 5.99i)8-s + (0.309 + 0.951i)9-s + 2.47·10-s + (−1.75 + 2.81i)11-s − 5.68·12-s + (0.309 + 0.951i)13-s + (−2.78 − 2.02i)14-s + (0.723 − 0.525i)15-s + (5.23 − 16.0i)16-s + (−1.40 + 4.32i)17-s + ⋯
L(s)  = 1  + (0.605 + 1.86i)2-s + (0.467 + 0.339i)3-s + (−2.29 + 1.67i)4-s + (0.123 − 0.380i)5-s + (−0.349 + 1.07i)6-s + (−0.379 + 0.275i)7-s + (−2.91 − 2.12i)8-s + (0.103 + 0.317i)9-s + 0.784·10-s + (−0.528 + 0.848i)11-s − 1.64·12-s + (0.0857 + 0.263i)13-s + (−0.743 − 0.539i)14-s + (0.186 − 0.135i)15-s + (1.30 − 4.02i)16-s + (−0.341 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502288 - 1.48181i\)
\(L(\frac12)\) \(\approx\) \(0.502288 - 1.48181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.75 - 2.81i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.856 - 2.63i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.276 + 0.850i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.00 - 0.728i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (1.40 - 4.32i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.60 + 2.62i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.00T + 23T^{2} \)
29 \( 1 + (-4.78 + 3.47i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.24 - 6.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.13 + 2.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.63 - 2.64i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.53T + 43T^{2} \)
47 \( 1 + (0.0565 + 0.0411i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.32 - 4.08i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.59 + 1.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.62 + 8.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + (-1.59 + 4.91i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.397 - 0.288i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.39 + 10.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.44 + 13.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (-0.837 - 2.57i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.29630969865224653445742292433, −10.54611032178660891452274977673, −9.278772552800084121912321952476, −8.833307919338075712204251975012, −7.941047746639194672847163213483, −6.94776683982051777870384938278, −6.17481693867146463853633256709, −4.93177051983817234821148338126, −4.42663381093639065118482046732, −3.02195957564277212995793943412, 0.812252771356833162329737704636, 2.54558757671163586290640327778, 3.10558802181483293509458457699, 4.30530404836266656628692666730, 5.46547498256907869297971179404, 6.65644469706050247496863323672, 8.303615405414891869839709665158, 9.100589834792577417871416921445, 10.07462577548902423050477487419, 10.73446408593533042797013610380

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