Properties

Label 2-429-429.194-c1-0-1
Degree $2$
Conductor $429$
Sign $-0.506 + 0.862i$
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  + (−0.618 + 0.850i)2-s + (−0.535 + 1.64i)3-s + (0.276 + 0.850i)4-s + (−0.319 + 0.232i)5-s + (−1.07 − 1.47i)6-s + (−2.89 − 0.940i)8-s + (−2.42 − 1.76i)9-s − 0.415i·10-s + (−2.36 − 2.32i)11-s − 1.54·12-s + (−2.11 + 2.91i)13-s + (−0.211 − 0.651i)15-s + (1.14 − 0.830i)16-s + (3.00 − 0.975i)18-s + (−0.285 − 0.207i)20-s + ⋯
L(s)  = 1  + (−0.437 + 0.601i)2-s + (−0.309 + 0.951i)3-s + (0.138 + 0.425i)4-s + (−0.143 + 0.103i)5-s + (−0.437 − 0.601i)6-s + (−1.02 − 0.332i)8-s + (−0.809 − 0.587i)9-s − 0.131i·10-s + (−0.714 − 0.700i)11-s − 0.446·12-s + (−0.587 + 0.809i)13-s + (−0.0546 − 0.168i)15-s + (0.285 − 0.207i)16-s + (0.707 − 0.229i)18-s + (−0.0639 − 0.0464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.506 + 0.862i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190000 - 0.331873i\)
\(L(\frac12)\) \(\approx\) \(0.190000 - 0.331873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.535 - 1.64i)T \)
11 \( 1 + (2.36 + 2.32i)T \)
13 \( 1 + (2.11 - 2.91i)T \)
good2 \( 1 + (0.618 - 0.850i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.319 - 0.232i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (7.44 + 2.41i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.0553iT - 43T^{2} \)
47 \( 1 + (4.15 - 12.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.75 - 11.5i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.21 + 11.3i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (3.98 - 2.89i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.26 - 4.49i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.64 - 13.2i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−11.59737748242168626632581844938, −10.86447843040927547475104762119, −9.777378170565245454797026299374, −9.030288601836306063895584800793, −8.185598805181343375127464227063, −7.19991029369695974137747822989, −6.17824602383755326924164507885, −5.16586556997294746591170803862, −3.89378144577043844424561701361, −2.84351814717762665259778731406, 0.27178871789880816283930673781, 1.86710860337407126779853490304, 2.88130513655555359591128652026, 4.93508389939457458717824115402, 5.80576777748751759234376143774, 6.88286983210130098149522531929, 7.83718380880950620286942508066, 8.706507770335964647355845698171, 10.00618684574458267547104090846, 10.46935418484614564927380202743

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