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Properties

Label 2-429-13.3-c1-0-2
Degree $2$
Conductor $429$
Sign $-0.646 + 0.763i$
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  + (1.36 + 2.36i)2-s + (−0.5 − 0.866i)3-s + (−2.71 + 4.70i)4-s − 2.20·5-s + (1.36 − 2.36i)6-s + (−0.0642 + 0.111i)7-s − 9.36·8-s + (−0.499 + 0.866i)9-s + (−3.00 − 5.20i)10-s + (−0.5 − 0.866i)11-s + 5.43·12-s + (−2.97 + 2.03i)13-s − 0.350·14-s + (1.10 + 1.90i)15-s + (−7.33 − 12.6i)16-s + (−1.15 + 1.99i)17-s + ⋯
L(s)  = 1  + (0.963 + 1.66i)2-s + (−0.288 − 0.499i)3-s + (−1.35 + 2.35i)4-s − 0.985·5-s + (0.556 − 0.963i)6-s + (−0.0242 + 0.0420i)7-s − 3.31·8-s + (−0.166 + 0.288i)9-s + (−0.949 − 1.64i)10-s + (−0.150 − 0.261i)11-s + 1.56·12-s + (−0.825 + 0.564i)13-s − 0.0936·14-s + (0.284 + 0.492i)15-s + (−1.83 − 3.17i)16-s + (−0.279 + 0.483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.646 + 0.763i$
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: \((2,\ 429,\ (\ :1/2),\ -0.646 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370210 - 0.798720i\)
\(L(\frac12)\) \(\approx\) \(0.370210 - 0.798720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.97 - 2.03i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.20T + 5T^{2} \)
7 \( 1 + (0.0642 - 0.111i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (1.15 - 1.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.427 - 0.740i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.17 - 2.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 2.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-3.53 - 6.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.54 - 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-2.83 + 4.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.74 - 4.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.80 - 13.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 - 5.70T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + (-3.32 - 5.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.842 + 1.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.03166364264589818998022956288, −11.36421865416033930442144876400, −9.608221413703703368555501079719, −8.217446878502714315931972766980, −7.996867035040780245849018924715, −6.86694467930568644764768463008, −6.33630812711942931514549057916, −5.07287962977279031683026358055, −4.34143852665223570427975646985, −3.11403182256135199320958933298, 0.41696832618832664302024657332, 2.48566721106877562463641242228, 3.51792652057883977293303370000, 4.54035819829468019793598626240, 5.07877076030471905296370366398, 6.44473150226683266133156492413, 8.026855792705702485471061535327, 9.303310910014746953594246659501, 10.10218838066753784481996351254, 10.75961180882312956007808001934

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