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Properties

Label 2-429-11.4-c1-0-9
Degree $2$
Conductor $429$
Sign $-0.348 - 0.937i$
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  + (1.89 + 1.37i)2-s + (0.309 − 0.951i)3-s + (1.07 + 3.30i)4-s + (−2.78 + 2.02i)5-s + (1.89 − 1.37i)6-s + (0.597 + 1.83i)7-s + (−1.06 + 3.28i)8-s + (−0.809 − 0.587i)9-s − 8.04·10-s + (1.08 + 3.13i)11-s + 3.47·12-s + (0.809 + 0.587i)13-s + (−1.39 + 4.30i)14-s + (1.06 + 3.27i)15-s + (−0.912 + 0.662i)16-s + (−0.819 + 0.595i)17-s + ⋯
L(s)  = 1  + (1.33 + 0.972i)2-s + (0.178 − 0.549i)3-s + (0.536 + 1.65i)4-s + (−1.24 + 0.904i)5-s + (0.772 − 0.561i)6-s + (0.225 + 0.695i)7-s + (−0.377 + 1.16i)8-s + (−0.269 − 0.195i)9-s − 2.54·10-s + (0.326 + 0.945i)11-s + 1.00·12-s + (0.224 + 0.163i)13-s + (−0.373 + 1.15i)14-s + (0.274 + 0.844i)15-s + (−0.228 + 0.165i)16-s + (−0.198 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41150 + 2.03151i\)
\(L(\frac12)\) \(\approx\) \(1.41150 + 2.03151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-1.08 - 3.13i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-1.89 - 1.37i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.78 - 2.02i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.597 - 1.83i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (0.819 - 0.595i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.885 + 2.72i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.86T + 23T^{2} \)
29 \( 1 + (2.15 + 6.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.95 + 4.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.16 + 6.67i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.89 - 5.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 + (3.09 - 9.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.84 + 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.127 - 0.392i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.65 + 4.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.32T + 67T^{2} \)
71 \( 1 + (5.57 - 4.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.99 - 12.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.94 + 7.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.56 + 1.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.22T + 89T^{2} \)
97 \( 1 + (9.69 + 7.04i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.56127202143354367234273426454, −11.19199719409925920885509955110, −9.381681454022027247635925191983, −8.183309309287183269142581107402, −7.27898716901527925659279463113, −6.94294236896019840471635030661, −5.83838547479443629736194364025, −4.64302831685269811249497300013, −3.70135776148159063213694584145, −2.59314803027755953357267419482, 1.15955227214728241903873086110, 3.30575465565242841428288672070, 3.77034205153099800363265112310, 4.77754902547905244058670108069, 5.44124871696241521347012456320, 7.08579895261167132717973258752, 8.335002134504343402011902859816, 9.086961063159891178938382895122, 10.60778123292007170433724261659, 11.02162036505191372146405789960

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