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Properties

Label 2-429-11.3-c1-0-9
Degree $2$
Conductor $429$
Sign $0.721 - 0.692i$
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  + (0.470 − 0.341i)2-s + (−0.309 − 0.951i)3-s + (−0.513 + 1.58i)4-s + (2.42 + 1.75i)5-s + (−0.470 − 0.341i)6-s + (0.122 − 0.378i)7-s + (0.657 + 2.02i)8-s + (−0.809 + 0.587i)9-s + 1.73·10-s + (−2.33 + 2.35i)11-s + 1.66·12-s + (−0.809 + 0.587i)13-s + (−0.0714 − 0.219i)14-s + (0.924 − 2.84i)15-s + (−1.68 − 1.22i)16-s + (1.52 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.332 − 0.241i)2-s + (−0.178 − 0.549i)3-s + (−0.256 + 0.790i)4-s + (1.08 + 0.786i)5-s + (−0.191 − 0.139i)6-s + (0.0464 − 0.143i)7-s + (0.232 + 0.715i)8-s + (−0.269 + 0.195i)9-s + 0.550·10-s + (−0.705 + 0.709i)11-s + 0.479·12-s + (−0.224 + 0.163i)13-s + (−0.0191 − 0.0587i)14-s + (0.238 − 0.734i)15-s + (−0.422 − 0.306i)16-s + (0.370 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52268 + 0.612037i\)
\(L(\frac12)\) \(\approx\) \(1.52268 + 0.612037i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 - 2.35i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.470 + 0.341i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.42 - 1.75i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.122 + 0.378i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-1.52 - 1.11i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.01 - 6.21i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.965T + 23T^{2} \)
29 \( 1 + (-3.31 + 10.2i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.13 + 5.90i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.07 - 3.29i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.84 - 8.76i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.74T + 43T^{2} \)
47 \( 1 + (3.36 + 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.32 + 5.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.71 + 8.35i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.61 + 1.90i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (6.50 + 4.72i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.85 - 14.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.88 + 2.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.31 + 5.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (8.64 - 6.27i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.57525887311114169769202899786, −10.17000963500668727675837588673, −9.876174326761804839749040880129, −8.249469720925955282154373184156, −7.64416259197463532955681795404, −6.54248312717917654849258290367, −5.62588031475126950491213777310, −4.39518943125298408219506782639, −2.94100213010387804669497639915, −2.03890138370247933436367182706, 1.04486280305918030231481144238, 2.89912032533200426671853034444, 4.71426081695431958407178366366, 5.23723807879786052015536240485, 5.91479318559231520226961599090, 7.09278315198361297824014289676, 8.813559305363462703457537087502, 9.138887030285184252680116553041, 10.27767485707329056991661573453, 10.66393402133382043546297009355

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