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Properties

Label 2-429-143.21-c1-0-7
Degree $2$
Conductor $429$
Sign $-0.610 - 0.791i$
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  + (−1.10 + 1.10i)2-s + 3-s − 0.441i·4-s + (1.95 + 1.95i)5-s + (−1.10 + 1.10i)6-s + (1.11 + 1.11i)7-s + (−1.72 − 1.72i)8-s + 9-s − 4.32·10-s + (−2.92 + 1.55i)11-s − 0.441i·12-s + (1.60 + 3.22i)13-s − 2.46·14-s + (1.95 + 1.95i)15-s + 4.68·16-s − 0.0613·17-s + ⋯
L(s)  = 1  + (−0.781 + 0.781i)2-s + 0.577·3-s − 0.220i·4-s + (0.874 + 0.874i)5-s + (−0.451 + 0.451i)6-s + (0.421 + 0.421i)7-s + (−0.608 − 0.608i)8-s + 0.333·9-s − 1.36·10-s + (−0.883 + 0.469i)11-s − 0.127i·12-s + (0.446 + 0.894i)13-s − 0.658·14-s + (0.504 + 0.504i)15-s + 1.17·16-s − 0.0148·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.610 - 0.791i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.610 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548642 + 1.11569i\)
\(L(\frac12)\) \(\approx\) \(0.548642 + 1.11569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + (2.92 - 1.55i)T \)
13 \( 1 + (-1.60 - 3.22i)T \)
good2 \( 1 + (1.10 - 1.10i)T - 2iT^{2} \)
5 \( 1 + (-1.95 - 1.95i)T + 5iT^{2} \)
7 \( 1 + (-1.11 - 1.11i)T + 7iT^{2} \)
17 \( 1 + 0.0613T + 17T^{2} \)
19 \( 1 + (-1.53 + 1.53i)T - 19iT^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 - 5.40iT - 29T^{2} \)
31 \( 1 + (1.00 + 1.00i)T + 31iT^{2} \)
37 \( 1 + (1.41 - 1.41i)T - 37iT^{2} \)
41 \( 1 + (6.30 - 6.30i)T - 41iT^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 + (6.90 - 6.90i)T - 47iT^{2} \)
53 \( 1 + 6.29T + 53T^{2} \)
59 \( 1 + (-8.28 + 8.28i)T - 59iT^{2} \)
61 \( 1 + 9.57iT - 61T^{2} \)
67 \( 1 + (-2.56 - 2.56i)T + 67iT^{2} \)
71 \( 1 + (-2.26 - 2.26i)T + 71iT^{2} \)
73 \( 1 + (-3.20 - 3.20i)T + 73iT^{2} \)
79 \( 1 + 7.07iT - 79T^{2} \)
83 \( 1 + (-9.65 + 9.65i)T - 83iT^{2} \)
89 \( 1 + (-4.26 + 4.26i)T - 89iT^{2} \)
97 \( 1 + (-9.43 - 9.43i)T + 97iT^{2} \)
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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

−11.22708124569719161493231781450, −10.23595423542685284624819822119, −9.512087257381493601810122900952, −8.677555079371995040617457616577, −7.898460932193519986400186324613, −6.88215650525275035746898393350, −6.29454388024142135363057378849, −4.89681825889859840496524712681, −3.20135860792087423287036721387, −2.08225347171366487044610718294, 1.01579829708885399156367975280, 2.13586232206139202140796489217, 3.45623986434002054047504069010, 5.18871257501211120896132393129, 5.80873725988734235068088902462, 7.64766372476456589590991580240, 8.370019507493321840605508930268, 9.154159953035390149118283908363, 9.957758917735132858765628010180, 10.56852541346849790034449544006

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