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Properties

Label 2-429-143.21-c1-0-17
Degree $2$
Conductor $429$
Sign $0.961 + 0.276i$
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  + (−1.92 + 1.92i)2-s + 3-s − 5.37i·4-s + (−0.179 − 0.179i)5-s + (−1.92 + 1.92i)6-s + (−1.23 − 1.23i)7-s + (6.48 + 6.48i)8-s + 9-s + 0.689·10-s + (−3.29 + 0.354i)11-s − 5.37i·12-s + (−3.59 − 0.334i)13-s + 4.76·14-s + (−0.179 − 0.179i)15-s − 14.1·16-s + 5.18·17-s + ⋯
L(s)  = 1  + (−1.35 + 1.35i)2-s + 0.577·3-s − 2.68i·4-s + (−0.0802 − 0.0802i)5-s + (−0.784 + 0.784i)6-s + (−0.468 − 0.468i)7-s + (2.29 + 2.29i)8-s + 0.333·9-s + 0.218·10-s + (−0.994 + 0.106i)11-s − 1.55i·12-s + (−0.995 − 0.0927i)13-s + 1.27·14-s + (−0.0463 − 0.0463i)15-s − 3.53·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.961 + 0.276i$
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: \((2,\ 429,\ (\ :1/2),\ 0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599225 - 0.0844090i\)
\(L(\frac12)\) \(\approx\) \(0.599225 - 0.0844090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.29 - 0.354i)T \)
13 \( 1 + (3.59 + 0.334i)T \)
good2 \( 1 + (1.92 - 1.92i)T - 2iT^{2} \)
5 \( 1 + (0.179 + 0.179i)T + 5iT^{2} \)
7 \( 1 + (1.23 + 1.23i)T + 7iT^{2} \)
17 \( 1 - 5.18T + 17T^{2} \)
19 \( 1 + (-5.00 + 5.00i)T - 19iT^{2} \)
23 \( 1 + 2.91iT - 23T^{2} \)
29 \( 1 + 9.44iT - 29T^{2} \)
31 \( 1 + (0.0134 + 0.0134i)T + 31iT^{2} \)
37 \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \)
41 \( 1 + (3.80 - 3.80i)T - 41iT^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + (0.296 - 0.296i)T - 47iT^{2} \)
53 \( 1 - 6.33T + 53T^{2} \)
59 \( 1 + (7.22 - 7.22i)T - 59iT^{2} \)
61 \( 1 - 2.44iT - 61T^{2} \)
67 \( 1 + (-6.64 - 6.64i)T + 67iT^{2} \)
71 \( 1 + (10.7 + 10.7i)T + 71iT^{2} \)
73 \( 1 + (-0.550 - 0.550i)T + 73iT^{2} \)
79 \( 1 + 0.139iT - 79T^{2} \)
83 \( 1 + (-2.55 + 2.55i)T - 83iT^{2} \)
89 \( 1 + (7.81 - 7.81i)T - 89iT^{2} \)
97 \( 1 + (-3.14 - 3.14i)T + 97iT^{2} \)
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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

−10.39193322955953644808619582486, −9.964348095943562311257631854081, −9.270105356536981527695268604989, −8.112859878968442077393257427813, −7.63068685741430372997457109231, −6.88194806062402868497815657727, −5.69703909566290278951245255975, −4.68163726600071396699475576930, −2.59469925645352806564141973958, −0.57740963002519923572432989946, 1.55020561612451118768064099376, 2.93890798409429214032990971869, 3.43419020170928181359296639017, 5.27340449697389216171260339425, 7.29149098078234693461024149216, 7.77933024891348083822389061253, 8.713453768333486389018857602937, 9.757098084976584155332313849097, 9.929834334876856287310846970155, 11.01001424919355279993947722329

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