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Properties

Label 2-429-33.32-c1-0-38
Degree $2$
Conductor $429$
Sign $0.951 + 0.308i$
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  + 2.48·2-s + (0.809 − 1.53i)3-s + 4.17·4-s + 2.08i·5-s + (2.01 − 3.80i)6-s + 0.856i·7-s + 5.41·8-s + (−1.69 − 2.47i)9-s + 5.18i·10-s + (−3.26 + 0.569i)11-s + (3.38 − 6.39i)12-s + i·13-s + 2.12i·14-s + (3.19 + 1.68i)15-s + 5.10·16-s − 3.81·17-s + ⋯
L(s)  = 1  + 1.75·2-s + (0.467 − 0.884i)3-s + 2.08·4-s + 0.932i·5-s + (0.821 − 1.55i)6-s + 0.323i·7-s + 1.91·8-s + (−0.563 − 0.826i)9-s + 1.63i·10-s + (−0.985 + 0.171i)11-s + (0.975 − 1.84i)12-s + 0.277i·13-s + 0.569i·14-s + (0.824 + 0.435i)15-s + 1.27·16-s − 0.926·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.84001 - 0.606628i\)
\(L(\frac12)\) \(\approx\) \(3.84001 - 0.606628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 1.53i)T \)
11 \( 1 + (3.26 - 0.569i)T \)
13 \( 1 - iT \)
good2 \( 1 - 2.48T + 2T^{2} \)
5 \( 1 - 2.08iT - 5T^{2} \)
7 \( 1 - 0.856iT - 7T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + 4.36iT - 19T^{2} \)
23 \( 1 + 3.17iT - 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 - 9.59T + 37T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 8.13iT - 43T^{2} \)
47 \( 1 + 4.07iT - 47T^{2} \)
53 \( 1 - 1.21iT - 53T^{2} \)
59 \( 1 - 4.04iT - 59T^{2} \)
61 \( 1 - 2.58iT - 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 7.49iT - 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−11.31519359834747435100492677176, −10.83392841915864370360733885152, −9.265270851110809298232068404770, −7.938971869347564352300496622601, −6.95316446604234409572677797035, −6.45998161605537439898142354527, −5.38353284469862651834265328120, −4.15994767037067129695505063170, −2.77460149879966330880918168013, −2.40654281026934387802672278023, 2.30144138317631446769365671020, 3.53466125090569710864235730510, 4.37992746020190659525899408871, 5.18897841628566599714023528816, 5.89069242683545064006754390417, 7.43104830464813703097690185909, 8.379744330752428114053104717570, 9.513836130533601510776429167546, 10.63702082588900394655859103128, 11.27350803788837043562099629207

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