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Properties

Label 2-429-11.5-c1-0-23
Degree $2$
Conductor $429$
Sign $-0.898 - 0.439i$
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  + (−0.311 − 0.960i)2-s + (−0.809 − 0.587i)3-s + (0.793 − 0.576i)4-s + (1.17 − 3.60i)5-s + (−0.311 + 0.960i)6-s + (−4.12 + 2.99i)7-s + (−2.43 − 1.76i)8-s + (0.309 + 0.951i)9-s − 3.83·10-s + (−3.30 − 0.313i)11-s − 0.980·12-s + (−0.309 − 0.951i)13-s + (4.16 + 3.02i)14-s + (−3.06 + 2.23i)15-s + (−0.332 + 1.02i)16-s + (−0.0488 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.220 − 0.678i)2-s + (−0.467 − 0.339i)3-s + (0.396 − 0.288i)4-s + (0.524 − 1.61i)5-s + (−0.127 + 0.391i)6-s + (−1.55 + 1.13i)7-s + (−0.860 − 0.625i)8-s + (0.103 + 0.317i)9-s − 1.21·10-s + (−0.995 − 0.0944i)11-s − 0.283·12-s + (−0.0857 − 0.263i)13-s + (1.11 + 0.809i)14-s + (−0.792 + 0.575i)15-s + (−0.0831 + 0.256i)16-s + (−0.0118 + 0.0364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.898 - 0.439i$
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: \((2,\ 429,\ (\ :1/2),\ -0.898 - 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155190 + 0.671054i\)
\(L(\frac12)\) \(\approx\) \(0.155190 + 0.671054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.313i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.311 + 0.960i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.17 + 3.60i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (4.12 - 2.99i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (0.0488 - 0.150i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.60 - 2.61i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 + (-4.22 + 3.06i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.32 + 7.15i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.97 - 3.61i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.58 + 3.33i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (-0.590 - 0.428i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.485 + 1.49i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.51 - 1.82i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.54 + 7.83i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.21T + 67T^{2} \)
71 \( 1 + (-3.33 + 10.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.05 + 2.94i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.51 - 4.67i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.43 + 4.42i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-4.73 - 14.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.51001044363129917334760683607, −9.705418772718227914629289823239, −9.207267334306128732028005159578, −8.128672578999021790563177880723, −6.59835633592915969711560419776, −5.72550742458163334684024886342, −5.20130732122396081536420520117, −3.16959545873780285634058698816, −2.00245414164375134999011697182, −0.45709233061481384678258018162, 2.85702457334744697210264442667, 3.39658369248171507661355547032, 5.31970966853056034759454367331, 6.51399331278877500752861461048, 6.88028545939313457314772081681, 7.46420659432413788701870346359, 9.102962553503810332285689520527, 10.17575280844681426437279836422, 10.52031186990637122547352246104, 11.37846347772319464990645593741

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