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Properties

Label 2-429-11.3-c1-0-10
Degree $2$
Conductor $429$
Sign $-0.00398 - 0.999i$
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  + (0.275 − 0.199i)2-s + (0.309 + 0.951i)3-s + (−0.582 + 1.79i)4-s + (3.18 + 2.31i)5-s + (0.275 + 0.199i)6-s + (−0.203 + 0.627i)7-s + (0.408 + 1.25i)8-s + (−0.809 + 0.587i)9-s + 1.34·10-s + (−0.0920 − 3.31i)11-s − 1.88·12-s + (−0.809 + 0.587i)13-s + (0.0693 + 0.213i)14-s + (−1.21 + 3.74i)15-s + (−2.68 − 1.95i)16-s + (−0.812 − 0.590i)17-s + ⋯
L(s)  = 1  + (0.194 − 0.141i)2-s + (0.178 + 0.549i)3-s + (−0.291 + 0.896i)4-s + (1.42 + 1.03i)5-s + (0.112 + 0.0816i)6-s + (−0.0770 + 0.237i)7-s + (0.144 + 0.444i)8-s + (−0.269 + 0.195i)9-s + 0.423·10-s + (−0.0277 − 0.999i)11-s − 0.543·12-s + (−0.224 + 0.163i)13-s + (0.0185 + 0.0570i)14-s + (−0.314 + 0.967i)15-s + (−0.671 − 0.487i)16-s + (−0.196 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00398 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00398 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.00398 - 0.999i$
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: \((2,\ 429,\ (\ :1/2),\ -0.00398 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28167 + 1.28678i\)
\(L(\frac12)\) \(\approx\) \(1.28167 + 1.28678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0920 + 3.31i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.275 + 0.199i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-3.18 - 2.31i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.203 - 0.627i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (0.812 + 0.590i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.57 + 4.84i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.10T + 23T^{2} \)
29 \( 1 + (0.247 - 0.762i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.03 + 2.92i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.206 + 0.637i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.545 + 1.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + (-1.78 - 5.50i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.61 - 6.98i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.106 - 0.328i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.47 + 2.52i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.12T + 67T^{2} \)
71 \( 1 + (7.84 + 5.69i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.20 + 9.85i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.15 + 3.74i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.22 - 4.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.75T + 89T^{2} \)
97 \( 1 + (-7.33 + 5.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.04416039999230149790193911824, −10.73905507386521386764501661349, −9.306387000092911801537720673173, −9.096426850438962995354010220889, −7.72999753348653532932728394923, −6.60744444528251415720014250270, −5.67951812028896843836189124769, −4.49541181735143746673101665449, −3.06611508656797982390788145225, −2.51514608015565367428419702312, 1.20266877200124199070807434524, 2.17912861242363216274344181522, 4.36808813584272027492435259620, 5.29278284568129284833717179312, 6.06469882721095441449052490799, 6.97907748854383952118153204506, 8.353100495930864041085139746813, 9.289818180965446049949230785341, 9.898100630413051268221270928926, 10.61853157400944816280464945354

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