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Properties

Label 2-429-143.18-c1-0-14
Degree $2$
Conductor $429$
Sign $-0.242 - 0.970i$
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  + (0.274 + 1.73i)2-s + (−0.309 + 0.951i)3-s + (−1.02 + 0.334i)4-s + (0.369 − 2.33i)5-s + (−1.73 − 0.274i)6-s + (3.03 + 1.54i)7-s + (0.732 + 1.43i)8-s + (−0.809 − 0.587i)9-s + 4.14·10-s + (1.89 − 2.71i)11-s − 1.08i·12-s + (3.43 + 1.11i)13-s + (−1.84 + 5.69i)14-s + (2.10 + 1.07i)15-s + (−4.03 + 2.93i)16-s + (0.415 − 0.302i)17-s + ⋯
L(s)  = 1  + (0.194 + 1.22i)2-s + (−0.178 + 0.549i)3-s + (−0.514 + 0.167i)4-s + (0.165 − 1.04i)5-s + (−0.707 − 0.112i)6-s + (1.14 + 0.584i)7-s + (0.258 + 0.508i)8-s + (−0.269 − 0.195i)9-s + 1.31·10-s + (0.572 − 0.820i)11-s − 0.312i·12-s + (0.951 + 0.307i)13-s + (−0.494 + 1.52i)14-s + (0.542 + 0.276i)15-s + (−1.00 + 0.733i)16-s + (0.100 − 0.0732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10205 + 1.41115i\)
\(L(\frac12)\) \(\approx\) \(1.10205 + 1.41115i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 + 2.71i)T \)
13 \( 1 + (-3.43 - 1.11i)T \)
good2 \( 1 + (-0.274 - 1.73i)T + (-1.90 + 0.618i)T^{2} \)
5 \( 1 + (-0.369 + 2.33i)T + (-4.75 - 1.54i)T^{2} \)
7 \( 1 + (-3.03 - 1.54i)T + (4.11 + 5.66i)T^{2} \)
17 \( 1 + (-0.415 + 0.302i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.71 - 3.42i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + 1.35iT - 23T^{2} \)
29 \( 1 + (-2.92 + 0.950i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.49 - 1.34i)T + (29.4 - 9.57i)T^{2} \)
37 \( 1 + (-0.470 + 0.923i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.36 + 3.24i)T + (24.0 - 33.1i)T^{2} \)
43 \( 1 + 3.09T + 43T^{2} \)
47 \( 1 + (0.800 - 0.407i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (10.0 + 7.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.30 - 8.43i)T + (-34.6 - 47.7i)T^{2} \)
61 \( 1 + (-5.16 - 7.10i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \)
71 \( 1 + (-1.62 + 10.2i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + (-1.03 - 0.528i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-2.38 + 3.28i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.473 - 0.0750i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (1.09 + 1.09i)T + 89iT^{2} \)
97 \( 1 + (-17.8 + 2.82i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.30075627593835460492486292475, −10.72512875926593324222921924712, −9.026241530922859864728468115151, −8.645349121882693184338718504208, −7.962332723436483734284755120221, −6.42967824972462483567858851453, −5.73231339339803142235132691011, −4.93105276800670422996957464079, −4.02908345655970638368386371401, −1.72823805775513467455284523411, 1.39956974710253997670281888134, 2.38069288854364186391301808773, 3.71236436320367996657033779847, 4.72061775059420479708182997606, 6.39256921603386020240857279416, 7.08133998266011593211262722819, 8.081991537917709630600727867074, 9.386588495923508643114887265188, 10.61951697576389569745801909140, 10.92716168178248286438353149232

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