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Properties

Label 2-429-11.9-c1-0-1
Degree $2$
Conductor $429$
Sign $-0.736 + 0.676i$
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  + (−0.711 + 2.19i)2-s + (−0.809 + 0.587i)3-s + (−2.67 − 1.94i)4-s + (0.686 + 2.11i)5-s + (−0.711 − 2.19i)6-s + (3.55 + 2.58i)7-s + (2.43 − 1.76i)8-s + (0.309 − 0.951i)9-s − 5.11·10-s + (−3.07 + 1.23i)11-s + 3.30·12-s + (−0.309 + 0.951i)13-s + (−8.18 + 5.94i)14-s + (−1.79 − 1.30i)15-s + (0.0974 + 0.299i)16-s + (2.04 + 6.30i)17-s + ⋯
L(s)  = 1  + (−0.503 + 1.54i)2-s + (−0.467 + 0.339i)3-s + (−1.33 − 0.971i)4-s + (0.306 + 0.944i)5-s + (−0.290 − 0.894i)6-s + (1.34 + 0.976i)7-s + (0.860 − 0.624i)8-s + (0.103 − 0.317i)9-s − 1.61·10-s + (−0.927 + 0.373i)11-s + 0.954·12-s + (−0.0857 + 0.263i)13-s + (−2.18 + 1.58i)14-s + (−0.464 − 0.337i)15-s + (0.0243 + 0.0749i)16-s + (0.497 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.736 + 0.676i$
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: \((2,\ 429,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322742 - 0.828999i\)
\(L(\frac12)\) \(\approx\) \(0.322742 - 0.828999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.07 - 1.23i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.711 - 2.19i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.686 - 2.11i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.55 - 2.58i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-2.04 - 6.30i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.326 + 0.237i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 + (8.35 + 6.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.50 + 7.70i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.37 + 3.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.56 + 4.03i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.91T + 43T^{2} \)
47 \( 1 + (-5.15 + 3.74i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.60 - 11.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.07 - 0.782i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.22 - 6.85i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + (-0.137 - 0.422i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.58 - 4.06i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.48 - 7.64i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.227 - 0.699i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (0.553 - 1.70i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.45190164314210792834010362497, −10.67608074252976549255180526951, −9.762653895756521850816511410593, −8.749254162643642797306710906702, −7.900888561265744222838980094374, −7.19900353808696384370997117650, −5.83731352044422491746484558670, −5.65825026627606758197957702417, −4.38801148514461801047440130510, −2.27690107935481449938544028572, 0.75564546730661491079184101566, 1.65339897353979718658854730033, 3.19605965594529286496156542204, 4.79342183377944948171150556822, 5.22670078158185786965959965484, 7.22683450882676606286994457322, 8.104207215072896763008321964884, 8.979111392028202760986456001021, 9.973778080923116657277236677232, 10.86068559960927826496819771811

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