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Properties

Label 2-429-143.109-c1-0-14
Degree $2$
Conductor $429$
Sign $-0.912 + 0.409i$
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  + (−1.43 − 1.43i)2-s + 3-s + 2.12i·4-s + (−2.46 + 2.46i)5-s + (−1.43 − 1.43i)6-s + (0.806 − 0.806i)7-s + (0.184 − 0.184i)8-s + 9-s + 7.09·10-s + (0.500 − 3.27i)11-s + 2.12i·12-s + (−0.994 − 3.46i)13-s − 2.31·14-s + (−2.46 + 2.46i)15-s + 3.72·16-s − 1.84·17-s + ⋯
L(s)  = 1  + (−1.01 − 1.01i)2-s + 0.577·3-s + 1.06i·4-s + (−1.10 + 1.10i)5-s + (−0.586 − 0.586i)6-s + (0.304 − 0.304i)7-s + (0.0653 − 0.0653i)8-s + 0.333·9-s + 2.24·10-s + (0.150 − 0.988i)11-s + 0.614i·12-s + (−0.275 − 0.961i)13-s − 0.619·14-s + (−0.637 + 0.637i)15-s + 0.931·16-s − 0.448·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.912 + 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112236 - 0.523818i\)
\(L(\frac12)\) \(\approx\) \(0.112236 - 0.523818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + (-0.500 + 3.27i)T \)
13 \( 1 + (0.994 + 3.46i)T \)
good2 \( 1 + (1.43 + 1.43i)T + 2iT^{2} \)
5 \( 1 + (2.46 - 2.46i)T - 5iT^{2} \)
7 \( 1 + (-0.806 + 0.806i)T - 7iT^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + (5.23 + 5.23i)T + 19iT^{2} \)
23 \( 1 + 8.45iT - 23T^{2} \)
29 \( 1 - 5.73iT - 29T^{2} \)
31 \( 1 + (-4.19 + 4.19i)T - 31iT^{2} \)
37 \( 1 + (-1.19 - 1.19i)T + 37iT^{2} \)
41 \( 1 + (1.04 + 1.04i)T + 41iT^{2} \)
43 \( 1 - 2.53T + 43T^{2} \)
47 \( 1 + (2.58 + 2.58i)T + 47iT^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 + (8.26 + 8.26i)T + 59iT^{2} \)
61 \( 1 + 2.26iT - 61T^{2} \)
67 \( 1 + (10.7 - 10.7i)T - 67iT^{2} \)
71 \( 1 + (-3.91 + 3.91i)T - 71iT^{2} \)
73 \( 1 + (7.89 - 7.89i)T - 73iT^{2} \)
79 \( 1 + 3.25iT - 79T^{2} \)
83 \( 1 + (2.19 + 2.19i)T + 83iT^{2} \)
89 \( 1 + (-11.1 - 11.1i)T + 89iT^{2} \)
97 \( 1 + (-2.64 + 2.64i)T - 97iT^{2} \)
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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

−10.86398024213903798046638763971, −10.17141033491802326740655898756, −8.801476649409060898402383668727, −8.346380396814733065601812058615, −7.45825605152807336976467017434, −6.39299084796769127301921753507, −4.41933726340357605964124851141, −3.19096215832144063333422116590, −2.53441584342265592327351078754, −0.44777317666177679440991133030, 1.67025369268776095936640513437, 3.89378697672065690426350121497, 4.76011665042640400492935395948, 6.22827301365270681485925467269, 7.41887730990144878845409774427, 7.892337368515399478288446133222, 8.751224905701945189427489400961, 9.252714074951383452192727747126, 10.19691471402025156091603115016, 11.76539805488492606882778100253

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