Properties

Label 2-429-429.428-c1-0-2
Degree $2$
Conductor $429$
Sign $0.882 + 0.469i$
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  + 2.73i·2-s + (−0.813 + 1.52i)3-s − 5.49·4-s + (−4.18 − 2.22i)6-s + 0.851·7-s − 9.55i·8-s + (−1.67 − 2.48i)9-s − 3.31i·11-s + (4.46 − 8.39i)12-s − 3.60·13-s + 2.33i·14-s + 15.1·16-s + (6.81 − 4.58i)18-s − 8.71·19-s + (−0.692 + 1.30i)21-s + 9.07·22-s + ⋯
L(s)  = 1  + 1.93i·2-s + (−0.469 + 0.882i)3-s − 2.74·4-s + (−1.70 − 0.909i)6-s + 0.321·7-s − 3.37i·8-s + (−0.558 − 0.829i)9-s − 1.00i·11-s + (1.29 − 2.42i)12-s − 1.00·13-s + 0.623i·14-s + 3.79·16-s + (1.60 − 1.08i)18-s − 1.99·19-s + (−0.151 + 0.284i)21-s + 1.93·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.882 + 0.469i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115627 - 0.0288498i\)
\(L(\frac12)\) \(\approx\) \(0.115627 - 0.0288498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.813 - 1.52i)T \)
11 \( 1 + 3.31iT \)
13 \( 1 + 3.60T \)
good2 \( 1 - 2.73iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 0.851T + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8.71T + 19T^{2} \)
23 \( 1 - 8.87iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 7.49iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 0.671iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.06105225629585745349174905776, −10.85672931108973935679231083522, −9.837723866989449823424273825398, −9.057586219152589030164742737816, −8.272363954659778385160317331893, −7.31367887876015349940804640199, −6.17539093940444220317573479247, −5.57681924593375342172460806433, −4.62649807274024841160591358179, −3.71778113766870582032869584778, 0.07877063027308025270644558124, 1.86521175740659871520923468378, 2.51476651640923192001494385426, 4.31878598861678942537632500469, 4.96533257109228075423794304769, 6.47013844736210560967151640720, 7.893356544307825903646893920188, 8.663159620456003292400367299604, 9.882262126481668275460664051118, 10.51037879226087502351616876076

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