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Properties

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

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

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