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Properties

Label 2-429-11.5-c1-0-15
Degree $2$
Conductor $429$
Sign $0.799 - 0.601i$
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  + (0.839 + 2.58i)2-s + (−0.809 − 0.587i)3-s + (−4.34 + 3.15i)4-s + (1.32 − 4.06i)5-s + (0.839 − 2.58i)6-s + (1.98 − 1.44i)7-s + (−7.41 − 5.38i)8-s + (0.309 + 0.951i)9-s + 11.6·10-s + (1.77 − 2.79i)11-s + 5.37·12-s + (−0.309 − 0.951i)13-s + (5.39 + 3.92i)14-s + (−3.45 + 2.51i)15-s + (4.36 − 13.4i)16-s + (−0.484 + 1.48i)17-s + ⋯
L(s)  = 1  + (0.593 + 1.82i)2-s + (−0.467 − 0.339i)3-s + (−2.17 + 1.57i)4-s + (0.590 − 1.81i)5-s + (0.342 − 1.05i)6-s + (0.751 − 0.545i)7-s + (−2.62 − 1.90i)8-s + (0.103 + 0.317i)9-s + 3.66·10-s + (0.536 − 0.844i)11-s + 1.55·12-s + (−0.0857 − 0.263i)13-s + (1.44 + 1.04i)14-s + (−0.892 + 0.648i)15-s + (1.09 − 3.35i)16-s + (−0.117 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.799 - 0.601i$
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: \((2,\ 429,\ (\ :1/2),\ 0.799 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48493 + 0.496201i\)
\(L(\frac12)\) \(\approx\) \(1.48493 + 0.496201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 + 2.79i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.839 - 2.58i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.32 + 4.06i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.98 + 1.44i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (0.484 - 1.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.65 - 1.92i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.86T + 23T^{2} \)
29 \( 1 + (-6.21 + 4.51i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.0504 - 0.155i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.67 - 4.12i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.10 - 2.25i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + (-6.21 - 4.51i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.00 + 3.10i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.22 + 2.34i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.27 - 3.92i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.64T + 67T^{2} \)
71 \( 1 + (1.62 - 4.99i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.0 - 8.76i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.95 + 6.01i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.42 + 7.46i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.06T + 89T^{2} \)
97 \( 1 + (-0.467 - 1.43i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.76419059272701860992742188078, −9.999174020173479424686280782174, −8.892852508912778160767617648244, −8.272817752182768519286246476917, −7.60855756186208632379672506135, −6.22009489166112377372651413008, −5.68762757498123898547108516746, −4.79779535408749818696603399052, −4.08840000751214576748757453576, −0.981854598753990236171297091906, 1.86660605848028138398103904602, 2.73904799346283200872425946575, 3.88859306991297420301104271062, 5.00560971136678815110188893199, 5.95309833833513140090841452609, 7.12561720681774273291684895717, 8.999175101054043043050229181681, 9.789564570637345655815759403740, 10.47646303410177861041585491923, 11.07218687460016344929135088894

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