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Properties

Label 2-429-13.4-c1-0-20
Degree $2$
Conductor $429$
Sign $-0.977 + 0.213i$
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  + (−0.458 − 0.264i)2-s + (−0.5 + 0.866i)3-s + (−0.860 − 1.48i)4-s − 3.29i·5-s + (0.458 − 0.264i)6-s + (0.609 − 0.351i)7-s + 1.96i·8-s + (−0.499 − 0.866i)9-s + (−0.871 + 1.51i)10-s + (0.866 + 0.5i)11-s + 1.72·12-s + (−3.53 + 0.723i)13-s − 0.372·14-s + (2.85 + 1.64i)15-s + (−1.19 + 2.07i)16-s + (−3.00 − 5.20i)17-s + ⋯
L(s)  = 1  + (−0.323 − 0.187i)2-s + (−0.288 + 0.499i)3-s + (−0.430 − 0.744i)4-s − 1.47i·5-s + (0.187 − 0.107i)6-s + (0.230 − 0.133i)7-s + 0.695i·8-s + (−0.166 − 0.288i)9-s + (−0.275 + 0.477i)10-s + (0.261 + 0.150i)11-s + 0.496·12-s + (−0.979 + 0.200i)13-s − 0.0995·14-s + (0.737 + 0.425i)15-s + (−0.299 + 0.519i)16-s + (−0.728 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.977 + 0.213i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.977 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0535793 - 0.497004i\)
\(L(\frac12)\) \(\approx\) \(0.0535793 - 0.497004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.53 - 0.723i)T \)
good2 \( 1 + (0.458 + 0.264i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.29iT - 5T^{2} \)
7 \( 1 + (-0.609 + 0.351i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (3.00 + 5.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 - 1.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.90 - 5.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.33 + 7.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.44iT - 31T^{2} \)
37 \( 1 + (8.39 + 4.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.15 + 3.55i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.37 + 7.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.76iT - 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + (7.96 - 4.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.55 - 9.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.27 - 3.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.44 + 4.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.06iT - 73T^{2} \)
79 \( 1 + 2.96T + 79T^{2} \)
83 \( 1 + 8.28iT - 83T^{2} \)
89 \( 1 + (-8.64 - 4.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.286 + 0.165i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48872504975941878943865470713, −9.790718686621061299600670253273, −9.037693821352003402295509486058, −8.451259851633351246709586075398, −7.02438985867947444405893658930, −5.52556752440448761411091406593, −4.96120529907733392346808788500, −4.17024082117373799233963946363, −1.91065247420580394484647209565, −0.35747933786277604920772270258, 2.32209903930311782533607273343, 3.48218936871255994211937540776, 4.78696188562822405693983913638, 6.49520049364697834247811103709, 6.78658478231273025111993217183, 7.941977404877647060867515784544, 8.568173295778990255373105139272, 9.914987433395167734501463374479, 10.65468480274849398733853791140, 11.56768558555699731867011094388

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