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Properties

Label 2-429-143.48-c1-0-15
Degree $2$
Conductor $429$
Sign $0.822 - 0.568i$
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  + (−0.769 + 0.342i)2-s + (0.669 + 0.743i)3-s + (−0.863 + 0.958i)4-s + (−0.666 + 0.483i)5-s + (−0.769 − 0.342i)6-s + (3.51 − 3.90i)7-s + (0.856 − 2.63i)8-s + (−0.104 + 0.994i)9-s + (0.346 − 0.600i)10-s + (3.11 + 1.14i)11-s − 1.29·12-s + (−0.177 − 3.60i)13-s + (−1.36 + 4.21i)14-s + (−0.805 − 0.171i)15-s + (−0.0257 − 0.244i)16-s + (5.47 + 2.43i)17-s + ⋯
L(s)  = 1  + (−0.544 + 0.242i)2-s + (0.386 + 0.429i)3-s + (−0.431 + 0.479i)4-s + (−0.297 + 0.216i)5-s + (−0.314 − 0.139i)6-s + (1.32 − 1.47i)7-s + (0.302 − 0.931i)8-s + (−0.0348 + 0.331i)9-s + (0.109 − 0.189i)10-s + (0.938 + 0.344i)11-s − 0.372·12-s + (−0.0491 − 0.998i)13-s + (−0.365 + 1.12i)14-s + (−0.207 − 0.0441i)15-s + (−0.00643 − 0.0611i)16-s + (1.32 + 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16707 + 0.364225i\)
\(L(\frac12)\) \(\approx\) \(1.16707 + 0.364225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-3.11 - 1.14i)T \)
13 \( 1 + (0.177 + 3.60i)T \)
good2 \( 1 + (0.769 - 0.342i)T + (1.33 - 1.48i)T^{2} \)
5 \( 1 + (0.666 - 0.483i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.51 + 3.90i)T + (-0.731 - 6.96i)T^{2} \)
17 \( 1 + (-5.47 - 2.43i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (0.00808 - 0.00171i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.127 + 0.0269i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (3.58 + 2.60i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-11.0 - 2.35i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (5.18 + 5.76i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (1.44 + 2.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.510 - 1.57i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.40 - 3.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.18 + 2.43i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-4.46 - 1.98i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (-0.836 + 1.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.63 - 4.29i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (2.34 + 7.21i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.46 - 1.06i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.36 - 6.07i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.03 - 5.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.25 - 11.9i)T + (-94.8 - 20.1i)T^{2} \)
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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

−11.05855218364961025708799309608, −10.15793901038802776872072121154, −9.523444317306279950286284382756, −8.212492815466862214964506329357, −7.79724051728374376956726734338, −7.13644514321268568817259661791, −5.33223913944080550628969372698, −4.04282687070875061836003019047, −3.64901658576857695336596827106, −1.24963381119138835139846262323, 1.31758301883181760147331868444, 2.38858467865776322708811214500, 4.29211785417457827465198638629, 5.30322438931759261308881260299, 6.30151393281129683684256852110, 7.86280231044397113085663245544, 8.481899822314167281272909089725, 9.088302532660850929382120015796, 9.914130097968364873457282652729, 11.40249231171321907641695280485

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