Properties

Label 2-429-13.3-c1-0-9
Degree $2$
Conductor $429$
Sign $-0.953 - 0.300i$
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  + (1.38 + 2.39i)2-s + (0.5 + 0.866i)3-s + (−2.83 + 4.91i)4-s + 3.07·5-s + (−1.38 + 2.39i)6-s + (1.16 − 2.02i)7-s − 10.1·8-s + (−0.499 + 0.866i)9-s + (4.26 + 7.37i)10-s + (0.5 + 0.866i)11-s − 5.66·12-s + (−3.02 − 1.95i)13-s + 6.46·14-s + (1.53 + 2.66i)15-s + (−8.40 − 14.5i)16-s + (−0.0407 + 0.0705i)17-s + ⋯
L(s)  = 1  + (0.979 + 1.69i)2-s + (0.288 + 0.499i)3-s + (−1.41 + 2.45i)4-s + 1.37·5-s + (−0.565 + 0.979i)6-s + (0.441 − 0.764i)7-s − 3.59·8-s + (−0.166 + 0.288i)9-s + (1.34 + 2.33i)10-s + (0.150 + 0.261i)11-s − 1.63·12-s + (−0.840 − 0.542i)13-s + 1.72·14-s + (0.397 + 0.687i)15-s + (−2.10 − 3.63i)16-s + (−0.00988 + 0.0171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406289 + 2.64356i\)
\(L(\frac12)\) \(\approx\) \(0.406289 + 2.64356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (3.02 + 1.95i)T \)
good2 \( 1 + (-1.38 - 2.39i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (0.0407 - 0.0705i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.19 - 2.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.746 - 1.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + (3.23 + 5.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.47 + 7.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.76T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + (4.09 - 7.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.94 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.50 - 4.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.99T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 0.656T + 83T^{2} \)
89 \( 1 + (4.64 + 8.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.373 - 0.646i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.89112557111736758646498833287, −10.43318409988004756376981537543, −9.414792027943456516142447818371, −8.755810446459410971128404038388, −7.43412812033798803845153639812, −7.00252330678476038887870313714, −5.61874168774453845146139980722, −5.15131749881976017322216139071, −4.12940292342387150303370093045, −2.78470158993158639501449318662, 1.58361589656972876604452661157, 2.23528958987429076405497817680, 3.34283938208491228895629729439, 4.89247784711249293122099600803, 5.60768602173260271553437267931, 6.48907224750077394763542501966, 8.450917175647153608412216721997, 9.475068274640019672199403432591, 9.887351755315796605854688508519, 10.92594335181507900797542449961

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