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Properties

Label 2-429-11.9-c1-0-23
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: $\chi_{429} (196, \cdot )$
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

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

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